https://lem12.uksw.edu.pl/api.php?action=feedcontributions&feedformat=atom&user=AndrzejSalwickiLem - Wkład użytkownika [pl]2026-08-23T03:54:50ZWkład użytkownikaMediaWiki 1.23.8https://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-20T17:48:22Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
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A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks <br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> ?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd numbers <br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math><br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in !<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
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-------------------------------------------------------------------- <br /><br />
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<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) </math>.<br />
------------------------------------------------------<br />
17 siepnia 2026<br />
I ta wersja też została odrzucona.<br />
----<br />
Nothing to share<br />
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tak brzmi recenzja lb raczej jej brak.<br />
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==== Wersja z 15 marca 2026 ====<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-20T09:21:53Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<br />
<br />
<br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks <br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> ?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd numbers <br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math><br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in !<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------<br />
17 siepnia 2026<br />
I ta wersja też została odrzucona.<br />
----<br />
Nothing to share<br />
-----<br />
tak brzmi recenzja lb raczej jej brak.<br />
-----------------------------------------------------------------------------<br />
<br />
==== Wersja z 15 marca 2026 ====<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-20T09:21:28Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<br />
<br />
<br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks <br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> ?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd numbers <br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math><br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in !<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
=============================<br />
17 siepnia 2026<br />
I ta wersja też została odrzucona.<br />
----<br />
Nothing to share<br />
-----<br />
tak brzmi recenzja lb raczej jej brak.<br />
-----------------------------------------------------------------------------<br />
<br />
==== Wersja z 15 marca 2026 ====<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-20T08:42:41Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<br />
<br />
<br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks <br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> ?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd numbers <br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math><br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in !<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
=============================<br />
17 siepnia 2026<br />
I ta wersja też została odrzucona.<br />
----<br />
Nothing to share<br />
-----<br />
tak brzmi recenzja lb raczej jej brak.<br />
=========================================================<br />
==== Wersja z 15 marca 2026 ====<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-20T08:41:04Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<br />
<br />
<br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks <br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> ?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd numbers <br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math><br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in !<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
=============================<br />
17 siepnia 2026<br />
I ta wersja też została odrzucona.<br />
----<br />
Nothing to share<br />
-----<br />
tak brzmi recenzja lb raczej jej brak.<br />
=========================================================<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-13T16:20:17Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<br />
<br />
<br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks <br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> ?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd numbers <br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math><br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in !<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-13T16:18:43Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<br />
<br />
<br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks <br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> ?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd numbers <br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in accordance with the lengths of their permutations. See also the trees <math>D_C</math> -it is a tree!<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T16:46:28Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<br />
<br />
<br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in accordance with the lengths of their permutations. See also the trees <math>D_C</math> -it is a tree!<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T16:45:43Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
}}<br />
<br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in accordance with the lengths of their permutations. See also the trees <math>D_C</math> -it is a tree!<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T16:26:01Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in accordance with the lengths of their permutations. See also the trees <math>D_C</math> -it is a tree!<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T16:25:02Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
|Nazwa=Quelques photos du massif du Vercors<br />
|wielkość=300<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in accordance with the lengths of their permutations. See also the trees <math>D_C</math> -it is a tree!<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T16:23:57Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
{{Galeria<br />
|Nazwa=Quelques photos du massif du Vercors<br />
|wielkość=300<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
<br />
}}<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in accordance with the lengths of their permutations. See also the trees <math>D_C</math> -it is a tree!<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T16:19:27Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png|Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png|the graph HC after permuting odd numbers in accordance with the lengths of their permutations. See also the trees <math>D_C</math> -it is a tree!<br />
| Plik:Computationforn27.png|Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T16:06:57Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria<br />
|Nazwa=Three not so easy remarks on Collatz conjecture<br />
| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after permuting odd numbers in accordance with the lengths of their permutations. See also the trees <math>D_C</math> -it is a tree!<br />
| Plik:Computationforn27.png |Computation for n equal 27 <br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T16:00:27Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=left<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
| Plik:Computationforn27.png |Computation for n=27<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T15:57:46Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria<br />
| Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
|pozycja=right<br />
|Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
|Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
|Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
| Plik:Computationforn27.png |Computation for n=27<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T15:56:12Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria<br />
Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a tree too, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
| Plik:Computationforn27.png |Computation for n=27<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T15:54:18Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture<br />
| wielkość=250<br />
| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
| Plik:Computationforn27.png |Computation for n=27<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T15:47:23Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
| Plik:Computationforn27.png |Computation for n=27<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T15:45:28Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
| Plik:Computationforn27.png |Example of computation for n=27<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T15:42:12Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | Trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png |Example of computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T15:38:52Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=350| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | Example of computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T13:08:44Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=350| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | example of computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T13:05:38Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three not so easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | example of computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:56:32Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three non-easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | example of computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
| Plik:Computationforn27.png | example of computation for n=27<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:53:44Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three non-easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | example of computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png| the graph HC after inverting edges -it is a tree!<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:48:06Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three non-easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | example of computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
}}<br />
<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:45:47Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three non-easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | example Computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
}}<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:38:36Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three non-easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | example Computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
}}<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
==== ABSTRACT ====<br />
<br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:36:51Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three non-easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:Computationforn27.png | example Computation for n=27<br />
| Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
}}<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:28:10Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
{{Galeria|Nazwa=Three non-easy remarks on Collatz conjecture| wielkość=250| pozycja=right<br />
| Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:19:50Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycja=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:16:38Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycjan=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:14:32Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycjan=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:13:34Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycjan=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz.png <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:12:06Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycjan=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC| Hotel Collatz.png <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:11:11Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycjan=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:07:26Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
A guide to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria|Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycjan=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-12T12:04:56Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<big>A guide</big> to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria||Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycjan=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/Szablon:GaleriaSzablon:Galeria2026-08-12T11:56:20Z<p>AndrzejSalwicki: Utworzono nową stronę "<includeonly><TemplateStyles src="Galeria/styles.css" /><div class="ImageGroup ImageGroup-{{{pozycja|right}}}" style="max-width:{{#expr: {{#if:{{{wielkość|}}}|{{{wie..."</p>
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{{Dokumentacja}}<br />
</noinclude></div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-11T07:09:42Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<big>A guide</big> to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria||Nazwa=Three easy remarks on Collatz conjecture| wielkość=250|pozycjan=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-11T07:01:05Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<big>A guide</big> to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{Galeria||Name=Three easy remarks on Collatz conjecture| size=250|position=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-11T06:56:50Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<big>A guide</big> to accept the thesis '''T1''' in six easy steps.<br />
<br />
<gallery>|Name=Three easy remarks on Collatz conjecture| size=250|position=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
</gallery><br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-11T06:53:27Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<big>A guide</big> to accept the thesis '''T1''' in six easy steps.<br />
<br />
{{gallery|Name=Three easy remarks on Collatz conjecture| size=250|position=right<br />
| Plik:Collatz_tree.png|Collatz tree - does it contain all natural numbers?<br />
| Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
| Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
}}<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/Plik:DrzewoHCpoModyfikacjach.pngPlik:DrzewoHCpoModyfikacjach.png2026-08-10T16:17:22Z<p>AndrzejSalwicki: </p>
<hr />
<div>=== This is the graph <math> HC </math> from another perspective===<br />
Yes! <br /><br />
# The odd numbers are permuted.<br />
# All edges go to the left.<br />
=== Ordering the odd numbers ===<br />
We are ordering the odd numbers with respect to the length of <math> 3n+1 </math> computations.<br />
Let <math> x </math>be a number of multiplications by 3, let <math> z </math> be the number of divisions by 2.We assume that the length of <math> 3n+1 </math> computation is <math> x+z </math>. <br /></div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/Plik:DrzewoHCpoModyfikacjach.pngPlik:DrzewoHCpoModyfikacjach.png2026-08-10T14:50:37Z<p>AndrzejSalwicki: </p>
<hr />
<div>=== Ordering the odd numbers ===<br />
We are ordering the odd numbers with respect to the length of <math> 3n+1 </math> computations.<br />
Let <math> x </math>be a number of multiplications by 3, let <math> z </math> be the number of divisions by 2.We assume that the length of <math> 3n+1 </math> computation is <math> x+z </math>. <br /></div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/CollatzCollatz2026-08-10T11:55:49Z<p>AndrzejSalwicki: </p>
<hr />
<div>== Recent version ==<br />
Dnia 4 sierpnia 2026<br /><br />
Przedstawiamy kolejną wersję artykułu.<br /><br />
Mamy nadzieję, że łatwiej będzie śledzić nasze argumenty.<br /><br />
Dodaliśmy kilka nowych rysunków.<br /><br />
I nowe ujecie dowodu tezy T1.<br /><br />
Miłego czytania [[https://lem12.uksw.edu.pl/images/4/41/CollatzConjectureBecomesTheorem.pdf]<br /><br />
------------------------------------------------------------------------------------------ <br /><br />
<big>A guide</big> to accept the thesis '''T1''' in six easy steps.<br />
<gallery><br />
Plik:Collatz_tree.png| 550px| Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
Plik:graphHC.png| Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
Plik:graphGoddnumbers.png | The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right? <br /><br />
</gallery><br />
Attention, please.<br />
<gallery><br />
Plik:TreesD-C.png | trees <math>D_C</math> i.e. left-down corners of the graph G<br />
Plik:Computationforn27.png | example Computation for n=27<br />
Plik:DrzewoHCpoModyfikacjach.png | the graph HC after inverting edges -it is a tree!<br />
</gallery><br />
* Fig. 2 page 6. Collatz tree -- how to assure that every natural number is a node of this tree?<br />
* Fig. 6 page 13. Hotel Collatz <math>\mathcal{HC}</math> -- every natural number is a node in this graph. But, is it a tree?<br />
* Fig.7 page 15. The graph <math>\mathcal{G}</math> of odd natural numbers with successors operation . If<math>\mathcal{G}</math> is a tree then the graph <math>\mathcal{HC}</math> is a treetoo, right?<br />
* Fig. 8 page 17. The sequence of trees <math>\mathcal{D}_C</math>. Do you recognize some similarity to te pairing function of G. Cantor?<br />
* Fig. 9 page 18. The path from 1 to 27 in the graph <math>\mathcal{G}</math>. Hence, the guest from the room no 27 learns how to reach his bed.<br />
* Fig. 10 page 20. The graph <math>\mathcal{G}</math> withall edges going to the left is a tree! <br />
Hence, all computations <math>3n+1 </math> are finite! Hence the graph <math>\mathcal{HC}</math> is atree<br />
-------------------------------------------------------------------- <br /><br />
<br />
<br />
ABSTRACT <br /><br />
as of Aug 4,2026 <br /><br />
<br />
We are presenting the paradox, i.e. two theses T1 and T2 that contradict each other. Third thesis T3 solves the problem.<br /><br />
<br />
'''T1'''. We show that the Collatz conjecture For every natural number ''n'' , the ''3n+1'' <br /> computation is finite is a ''semantically valid statement". <br />
The sufficient and necessary criterion <math>\varphi(n)</math> for termination of <math>3n+1</math> computation is given. <br /><br />
We prove that, every instance <math>\varphi (n/r)</math> of the criterion where the variable <math>n</math> is replaced by any natural number <math>r\neq0</math>, is ''a theorem of Peano's arithmetic'', Hence, the set <math>\left\lbrace \varphi(n/r)\right\rbrace _{r=0}^{\infty} \subset Th(\mathcal{PA})</math> is a recursive subset of the set of theorems. <br /><br />
'''T2'''. Paradoxically, the Collatz conjecture itself, '''is not a theorem''' of number theory (Peano's arithmetic or a similar elementary theory). <br />
It is so because, 1. the formula <math>\forall_{n}\varphi(n)</math> obtained by putting the general quantifier <math>\forall_{n}</math> in front of formula <math>\varphi(n)</math>, may obtain the value <math>\mathbf{\mathbb{F}}</math> = false, in a non-standard model of Peano's arithmetic \ and \ <br />
2. there is no way to bound the classical quantifier to the set of standard, reachable natural numbers. <br /><br />
To avoid the paradox, we will conduct our considerations in the formalized \textit{algorithmic} theory <math>\mathcal{ATN}</math> of natural numbers. The logical consequence operation of the theory is determined by the calculus of programs <math>\mathcal{AL}</math>, which is an extension of the predicate calculus. <br />
The halting condition of the Collatz computations is written as an algorithmic formula. <br /><br />
'''T3'''. We are '''proving''' that, four infinite sets <math>St_{0},St_{1},St_{2},St_{3}</math> of formulas, are the ''recursive sets'' of theorems of the theory <math>\mathcal{ATN}</math>. Hence, every formula of the set <math>St_{3}</math> has a proof. Making use of the infinitary inference rule <math>R_{3}</math> to the infinite set <math>St_{3}</math> of premises we conclude the proof of the Main theorem <br /><br />
<math>\begin{equation}<br />
\mathcal{ATN} \vdash \forall_{n> 0} \left( \underbrace{\left\{<br />
\begin{array}{l}<br />
q\leftarrow 1 ; \\<br />
\mathbf{while}\ n \neq q \ \mathbf{do}\\ <br />
\quad q\leftarrow q+1\\ \mathbf{od}<br />
\end{array} <br />
\right\}(n=q)}_{{\mathbb{IF}\ n \ is\ a\ natural\ number}} \implies <br />
\underbrace{\left\{ \begin{array}{l}<br />
m\leftarrow\rho(n); \\<br />
\mathbf{while}\ m\neq 1 \ \mathbf{do}\\ <br />
\quad m \leftarrow \rho(3m+1)\\ \mathbf{od}<br />
\end{array}<br />
\right\} (m=1)}_{\mathbb{THEN}\ the\ computation\ for\ n\ is\ finite\ \mathbb{FI} } \right) \qquad<br />
<br />
\end{equation}<br />
</math><br />
where <br /><br />
'''Definition.''' The function <math>\rho</math> is defined as <br />
<math> \[ \rho(n)=(2j+1) \Longleftrightarrow \exists_{i}\exists_{j}\,n=2^{i}\cdot (2j+1) \] </math>.<br />
------------------------------------------------------------ <br /><br />
Możemy ogłosić, że [http://arxiv.org/abs/2310.13035 dowód] hipotezy Collatza został ukończony. <br /><br />
Oto wersja złożona do druku [https://lem12.uksw.edu.pl/images/6/69/CollatzConjecturebecomesTheorem2026-03-15.pdf]<br /><br /><br />
I odrzucona. <br /><br />
Artykuł złożono '''15 marca 2026'''.<br /><br />
Redaktor naczeelny wyznaczył redaktora odpowiedzialnego za zasiegniecie opinii i podjęcie decyzji 16 marca o godzinie 20:52.<br /><br />
Redaktor odpowiedzialny podjął decyzję '''17 marca 2026 o godzinie 12:17'''.<br /><br />
Oto cała recenzja i decyzja<br /><br />
''Unfortunately, we cannot accept it for publication. The paper belongs to computer science and not mathematics. So the proper venue for its publication should be a computer science journal. It seems to me that the bare argument for the Collatz conjecture presented in your paper is not very complicated (this does not mean that I verified it). Why not extract it, omitting the programming jargon? Then it would be more accessible to mathematicians.''<br />
<br />
<br /><br />
<br />
<br />
<big><big>Absract</big></big> <br /><br />
We are showing that the following conjecture<br />
''For every <math>n</math>, if <math>n</math> is a natural number then Collatz computation is finite.'' <br />
is a semantically valid statement. <br /><br />
<br />
<br />
This is asserted by the Main lemma. <br /><br />
<br />
A corollary of the lemma says: every instance of the cnjecture where the variable <math>n</math> is replaced by any natural number <math>r \neq 0</math>, is a theorem of arithmetic, in which the addition is the only operation. <br /><br />
Note, the set <math>St_0</math>of is a recursive set of theorems of Presburger arithmetic, hence the theorems of algorithmic theory of natural numbers. <br /><br />
<br />
<br />
Paradoxically, the Collatz conjecture itself is not a theorem of number theory (Peano's arithmetic), nor any mathematical theory that uses the first-order language and the classical predicate logic. <br /><br />
<br />
It is so because, '''1''') there is no first-order theory such that all its models are isomorphic to the standard model of natural numbers <br />
and hence '''2''') the infinite computations can be n observed in a ''non-standard computable'' model of the elementary theory of natural numbers with addition. <br /><br />
<br />
To avoid the paradox, we will use the calculus of programs <math>\mathcal{AL}</math> instead of the predicate calculus. The halting condition <math>H</math> of the Collatz computations is written as an algorithmic formula. <br />
<br />
<math> \qquad \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {the\ computation\ for\ n\ is\ finite} } \qquad (H) <br />
</math> <br /><br />
or by another, equivalent formula with iteration quantifier instead of '''while''' <br /><br />
<br />
There is no finite, traditional proof the following theorem . <br/><br />
<math> \begin{array}{p{14cm}}<br />
% We are answering to the question (\textit{i}) formulating the thesis of the \textsc{Theorem}\eqref{thM}. <br />
% \label{main}<br />
\mathcal{ATN}\vdash <br />
\forall_{n \neq 0} \underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
q:=1; \\<br />
\mathbf{while}\ n \neq q \\<br />
\mathbf{do} \\<br />
\quad q:=q+1 \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(n=q) } }_{\color{black}{IF\ n>0 \ is\ a\ natural\ number\ }}<br />
\implies <br />
\underbrace{\boxed{\left\lbrace \begin{array}{l}<br />
m:=n \div 2^{\kappa(n)}; \\<br />
% (*\ \ m= 2^{\kappa(n)} (2 \rho(m)+1) \ \ *) \\<br />
\mathbf{while}\ m \neq 1 \\<br />
\mathbf{do} \\<br />
\quad m:=3 \cdot m +1; \\<br />
\quad m:= m \div 2^{\kappa(m)} \\<br />
\mathbf{od} <br />
\end{array}\right\rbrace(m=1)} }_{ {THEN\ the\ computation\ for\ n\ is\ finite\ FI} } <br />
\end{array} </math><br />
<br />
F<small>unction <math>\kappa</math> for a given natural number <math> n </math> returns the multiplicity of 2 in the factorization of the number <math>n</math>.</small> <br />
<br />
<br />
Instead, we are presenting an rgument showing that the proof can be carried out in the calculus of programs <math>\mathcal{AL}</math>. To achieve his goal one has to construct an infinite tree <math>\mathcal{D}</math>. The root of the tree is the halting formula . The formula is the consequence of the infinitary inference rule <math>R_3</math> of the algorithmic logic <math>\mathcal{AL}</math>. For each premise one can construct a a finite subtree, i.e. a finite proof which is using one formula of the set <math>St_0</math>. \\<br />
Note, that that the set <math>St_0</math> is a recursive set of formulas without variables and that all its elements are theorems of Presburger's arithmetic.<br /> <br />
end of Abstract 01/10/2025<br />
<br />
==Introduction==<br />
Let's consider the statement<br/><br />
for every natural number <math>n</math>, the following program <math>Cl</math> has a finite computation.<br/><br />
<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} </math> <br /><br />
We begin by noting that the truth of the above statement entails the truth of Collatz's thesis as it was formulated before World War II. <br /><br />
But in 1937, neither computers nor programming languages existed.<br /><br />
On the other hand, the theory of algorithms did exist and was already well developed. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ...<br /><br />
In London, Alan Turing created the abstract Turing machine.<br /><br />
In Moscow, Kolmogorov and in Kazan, Maltsev explored the concept of a computable function.<br /><br />
<br /><br />
In Warsaw, Alfred Tarski, together with his students Mojżesz Presburger and Stanisław Jaskowski, obtained important results concerning the theory of addition of natural numbers.<br />
<br />
==Our observations from 2004==<br />
* The Collatz algorithm <math>Cl</math> does not require multiplication or division operations. Multiplying by 3 (because 3x=x+x+x) and dividing by 2 (a simple algorithm adding every other 1 is sufficient), is sufficient.<br />
* In the algebraic structure <math>\mathfrak{M}</math>, which is a non-standard model of the elementary theory of addition of natural numbers (there is one, see below), the algorithm <math>Cl</math> has an infinite computation for many arguments.<br />
* Therefore, the Collatz theorem cannot be proven based on the axioms of the elementary theory of addition of natural numbers.<br />
* Moreover, in the language of elementary theory of addition, there is no stopping formula for the Collatz algorithm! It is a corollary from the Goedel incompleteness theorem. <br /><br />
So what do we have to prove?<br />
<br />
<br />
==Have a look==<br />
[ [File:https://lem12.uksw.edu.pl/wiki/Plik:Collatz_tree.png ]]<br />
<br />
==Correct formulation of the Collatz theorem==<br />
In the standard structure of natural numbers with the addition operation,<br />
our program <math>Cl</math> has a finite computation for each argument ''n''.<br />
<br />
==Stop formula==<br />
i.e.<br />
=== A necessary and sufficient condition for the computation to be finite===<br />
Therefore, we need to create a formula <math>\theta</math> (a logical expression) such that it evaluates to true if and only if the computation of the program <math>Cl</math> is finite. There are many such formulas in the language of program calculation, i.e. algorithmic logic.<br/><br />
,<math>\qquad \theta:\,\left\{\begin{array}{l} \mathbf{while}\ n \neq 0 \ \mathbf{do} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{od} \end{array}\right\} (n=1) </math><br />
<br /><br />
The value of the <math>\theta</math> formula depends only on the initial value of the "n" variable. This formula is satisfied by the value of the variable "n" if and only if the evaluation of the while ... program is finished and the final value of the variable "n" is equal to 1. <br /><br />
Other formulas can also be considered, e.g., <br /><br />
,<math>\qquad \xi:\,\bigcup \left\{\overbrace{\begin{array}{l} \mathbf{if}\ n \neq 0 \ \mathbf{then} \\<br />
\quad \mathbf{if}\ odd(n) \ \mathbf{then}\ n:=3n+1 \ \mathbf{else}\ n:=n/2\ \mathbf{fi} \\<br />
\mathbf{fi} \end{array} }^{K}\right\} (n=1) </math> <br /><br />
{co reads: "there exists an iteration <math>K^i</math> of the program <math>K</math> such that after executing <math>K^i</math> the equality <math>n=1</math> is satisfied."} <br/><br />
In other words, we are dealing with an upper bound on the values of the formulas <math>K^i(n=1)</math>, where <math>i= 0,1,2 \dots</math>.<br /><br />
<br />
The second part of the problem is much more difficult: we must prove the stopping formula using the axioms of program calculus and the axioms of the algorithmic theory of natural numbers.<br/><br />
<br />
<br /><br />
==Elementary Theory of Addition of Natural Numbers==<br />
The previous observation that Collatz's theorem cannot be proved in this theory remains valid. However, the properties of the non-standard model of this theory and a few of its theorems will be helpful in further considerations.<br /><br />
<br /><br />
This theory is defined by specifying three components:<br />
* the language,<br />
* the logic, i.e., the consequence operation, and <br />
* the axioms specific to this theory.<br /><br />
'''Language.''' The expressions of the language are composed of the following symbols: variable symbols, e.g., x, y, n, the + symbol for a binary operation, the = symbol for a binary relation, constant symbols, logical functor symbols, and auxiliary symbols, e.g., parentheses.<br /><br />
. Examples of expressions are...<br /><br />
<br />
'''Logic.''' The consequence (inference) operation is determined by specifying the axioms of first-order logic and the rules of inference.<br /><br />
'''Axioms.'''<br />
<math><br />
\begin{align} <br />
\tag{a} \forall_x\ x+1 &\neq 0 \\<br />
\tag{b} \forall_x\, \forall_y\ x+1=y+1 &\implies x=y \\<br />
\tag{c} \forall_{x}\ x+0&=x \\<br />
\tag{d} \forall_{x,y}\ (y+1)+x&=(y+x)+1 \\<br />
\tag{I} \Phi(0)\land \forall_x\,[\Phi(x) \implies \Phi(x+1)]&\implies \forall_x\Phi(x) <br />
\end{align} <br />
</math><br />
The expression <math>\Phi(x)</math> may be replaced by any formula. The result is an axiom of theory <br />
This is the induction scheme. <br /><br />
We augment the set of axioms adding four axioms that define a coiple of useful notions. <br /><br />
<math><br />
\begin{align}<br />
\tag{e} even(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y \\<br />
% \tag{o} odd(x) &\stackrel{df}{\equiv} \exists_y\, x=y+y+1 \\<br />
% \tag{D2} x\, div\, 2 = y &\equiv (x=y+y\, \lor\, x=y+y+1) \\<br />
% \tag{3x} 3x&\stackrel{df}{=} x+x+x<br />
\end{align} <br />
</math><br />
<br />
'''Models of Presburger Arithmetic'''<br /><br />
As expected, the sequence of standard values 0, 1, 2, 3, ... is a model of this theory.<br />
<br />
Stanisław Jaśkowski discovered another, nonstandard model of Presburger arithmetic in 1929.<br />
<br />
[[File:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]<br />
The universe of the model is a subset of the set of complex numbers <math>a+\math b</math><br />
where <math>a \in \mathbb{Z} </math> i.e. a is an integer number and <math>b \in \mathbb{Q}^+ </math> is a positive rational number. Additionally, whenever <math>b=0 </math> we have <math>a>0</math>.<br />
Addition is defined as usual addition of complex numbers.<br />
<br />
Both models are computable. There are also unpredictable models with arbitrarily high power.<br />
<br />
==Algorithmic Theory of Natural Numbers==<br />
* Language. The alphabet of a language contains a set of variables, e.g., x,y. a functor + a two-argument addition operation, two constants 0 and 1, a relation sign = equality.<br /><br />
Terms (i.e., nomenclature expressions): this is the smallest set of expressions containing variables, constants, and closed under the combination of two terms in this way (t1 + t2).<br /><br />
Formulae.<br />
* Logic. Program calculus. Program calculus includes first-order logic. In addition to first-order formulas, the language of program calculus also contains algorithmic formulas. The simplest such formula is a string consisting of a program and a formula (usually a first-order formula) following it.<br />
To the axioms of first-order logic, axioms describing the properties of program-generating connectives should be added; see [[Algorithmic Logic]].<br />
To the inference rules of first-order logic, rules specific to program calculus should be added.<br /><br />
<br />
* Axioms of the theory.<br /><br />
Only three formulas.<br /> <br />
<math><br />
\begin{eqnarray}<br />
\tag{ATN1} \forall_x\, x+1 \neq 0 &&\\<br />
\tag{ATN2} \forall_{x,y}\,x+1=y+1 \implies x=y &&\\<br />
\tag{ATN3}\forall_x\, \{y :=0; \mathbf{while}\ y\neq x\ \mathbf{do}\ y:=y+1\ \mathbf{od} \}\,(y=x) &&<br />
\end{eqnarray} </math><br /><br />
<br />
These are essentially the axioms of the successor theory.<br/><br />
The ATN1 formula states that 0 is not the successor of any natural number.<br/><br />
The ATN2 formula states that the successor is a one-to-one function.<br/><br />
The formula ATN3 states that every natural number is ''reachable'' from zero by adding a finite number of ones.<br/><br />
In this theory, one can write definitions for addition, multiplication, and any computable function.<br />
<br />
==Analiza formuły stopu==<br />
xxx<br />
<br />
==Trójki ==<br />
Spostrzeżenie (wynikłe z przygladania się formule stopu).<br /><br />
<br />
<math>\forall_{n \neq 0} \exists_{x,y,z}\ n \cdot 3^x+y=2^z </math><br />
<br />
==Drzewo Collatza==<br />
[[Plik:StratDrzewoCollatza.png|thumb|center |750px| Rys. 1 Fragmenty warstw <math>W_0, \dots W_4 </math> drzewa Collatza ]]<br />
<br />
==Własności obliczeń na trójkach==<br />
Tutaj napiszemy więcej<br /><br />
==Kalejdoskop==<br />
<br />
Oglądaj rysunki, wykonuj obliczenia, rozwiązuj zadania, formułuj swoje zdanie, próbuj je uzasadnić, ...<br /><br />
<br />
Tu znajdziesz ....<br /><br />
===Obliczenia utemperowane===<br />
[[Plik:ObliczN19.pdf.png|thumb|center|750px|Utemperowane obliczenie dla n=76]]<br />
Trzy zadania. Odpowiedz czy są one jakos powiązane?<br /><br />
* Masz do dyspozycji bardzo wiele trójkątnych płytek, w dwu kolorach. <br />
Czy potrafisz ułożyć chodnik łączący posesje o numerze n z numerem 1?<br />
*[[Ułamek piętrowy]]<br />
* Czy obliczenie 3x+1 jest skończone dla każdej liczby naturalnej?<br />
<br />
===Struktury algebraiczne===<br />
Struktura liczb naturalnych. <br /><br />
Algebra Jaśkowskiego.<br /><br />
===Teorie===<br />
elementarna teoria liczb naturalnych z dodawaniem.<br /><br />
algorytmiczna teoria liczb naturalnych<br /><br />
===Zadania===<br />
<br />
<br />
<math>\alpha</math><br /><br />
<br />
== Archiwum kolejnych wersji pracy ==<br />
[CollatzConjecturebecomesTheorem11Aug23 http://lem12.uksw.edu.pl/images/3/3b/CollatzConjecturebecomesTheorem11Aug23.pdf]<br />
<br />
[https://dx.doi.org/10.2139/ssrn.4158238 \On Collatz theorem II.pdf wersja z 5 czerwca 2022 ]<br />
<br />
][http://lem12.uksw.edu.pl/images/a/ab/On-Collatz-thm17-09-21.pdf wersja z 20 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/7/7d/Algorytmy-bliskie-Collatzowi.pdf algorytmy wokół Collatzowe]<br />
<br />
[http://lem12.uksw.edu.pl/images/c/c0/On-Collatz-thm-27-09-21.pdf wersja z 27 wrzesnia 2021]<br />
<br />
[http://lem12.uksw.edu.pl/images/8/8f/On-Collatz-thm-7-10-21.pdf wersja z 7 pażdziernika 2021]</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/Plik:GraphHC.pngPlik:GraphHC.png2026-08-10T11:52:34Z<p>AndrzejSalwicki: </p>
<hr />
<div>=== Lemma ===<br />
The graph <math>HC</math> is a tree '''iff''' the tree Cl contains all natural numbers.<br />
-----</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/Plik:GraphHC.pngPlik:GraphHC.png2026-08-10T11:51:56Z<p>AndrzejSalwicki: </p>
<hr />
<div>=== Lemma ===<br />
The graph <math>HC</math> is a tree '''iff''' the tree Cl contains all natural numbers.</div>AndrzejSalwickihttps://lem12.uksw.edu.pl/wiki/Plik:GraphGoddnumbers.pngPlik:GraphGoddnumbers.png2026-08-10T11:47:01Z<p>AndrzejSalwicki: </p>
<hr />
<div>=== Definition of the graph G ===<br />
<math> \color{blue}G=\langle V,E \rangle </math><br /><br />
<br />
The set of all odd natural numbers constitutes the set <math>V</math>.<br /><br />
<br />
A pair <math>\langle n,m\rangle </math> of odd numbers is an edge iff <math>n</math> is indivisible by 3 and there exists the natural number <math>i>0</math> such that <math>\color{blue} m=\frac{n\cdot 2^{2i-(n \mod 3)-1}-1}{3}</math><br />
=== Lemma ===<br />
The graph <math>G</math> is a tree iff the graph <math>HC</math> is a tree.<br />
-----</div>AndrzejSalwicki