Collatz theorem - Historia wersji

AndrzejSalwicki: /* Elementary theory od addition of natural numbers and its models */

2022-09-28T12:39:12Z

‎Elementary theory od addition of natural numbers and its models

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2022-09-28T10:22:33Z

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2022-09-28T10:21:36Z

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2022-09-28T10:20:07Z

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2022-09-28T10:19:12Z

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2022-09-28T10:17:40Z

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2022-09-28T10:10:32Z

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2022-09-28T10:07:21Z

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AndrzejSalwicki o 09:54, 28 wrz 2022

2022-09-28T09:54:59Z

AndrzejSalwicki: /* Elementary theory od addition of natural numbers and its models */

2022-09-27T07:30:09Z

‎Elementary theory od addition of natural numbers and its models

← poprzednia wersja		Wersja z 12:39, 28 wrz 2022
Linia 57:		Linia 57:


−	Nonstandard model of Presburger arithmetic was invented by Stanisław Jaśkowski in 1929.	+	<big>Nonstandard model of Presburger arithmetic</big> was invented by Stanisław Jaśkowski in 1929.

	[[Plik:MonStandardModel.png\|center\|thumb\|600px\|Nonstandard model of Presburger arithmetic]]		[[Plik:MonStandardModel.png\|center\|thumb\|600px\|Nonstandard model of Presburger arithmetic]]

@@ Linia 9: / Linia 9: @@
 Consider the following sentence<br/>
 for every natural number  <math>n</math>, the program  <math>Cl </math> has the finite computation<br/>
-<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l}  \mathbf{while}\ n \neq 0 \ \mathbf{do} \\
+<math>\qquad Cl:\,\left\{\begin{array}{l}  \mathbf{while}\ n \neq 0 \ \mathbf{do} \\
 \quad \mathbf{if}\  odd(n) \ \mathbf{then}\  n:=3n+1 \  \mathbf{else}\ n:=n/2\  \mathbf{fi} \\
-\mathbf{od} \end{array}\right\} \color{blue}</math> <br />
+\mathbf{od} \end{array}\right\} </math> <br />
 Note,  the truthfulness of the above sentence implies the truthfulness of Collatz's thesis, as it was formulated before World War II, in 1937.. <br />
 But then there were no computers or computer programming languages<br />

@@ Linia 9: / Linia 9: @@
 Consider the following sentence<br/>
 for every natural number  <math>n</math>, the program  <math>Cl </math> has the finite computation<br/>
-\color{blue}<math>\qquad Cl:\,\left\{\begin{array}{l}  \mathbf{while}\ n \neq 0 \ \mathbf{do} \\
+<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l}  \mathbf{while}\ n \neq 0 \ \mathbf{do} \\
 \quad \mathbf{if}\  odd(n) \ \mathbf{then}\  n:=3n+1 \  \mathbf{else}\ n:=n/2\  \mathbf{fi} \\
-\mathbf{od} \end{array}\right\} </math> <br />
+\mathbf{od} \end{array}\right\} \color{blue}</math> <br />
 Note,  the truthfulness of the above sentence implies the truthfulness of Collatz's thesis, as it was formulated before World War II, in 1937.. <br />
 But then there were no computers or computer programming languages<br />

@@ Linia 9: / Linia 9: @@
 Consider the following sentence<br/>
 for every natural number  <math>n</math>, the program  <math>Cl </math> has the finite computation<br/>
-<math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l}  \mathbf{while}\ n \neq 0 \ \mathbf{do} \\
+\color{blue}<math>\qquad Cl:\,\left\{\begin{array}{l}  \mathbf{while}\ n \neq 0 \ \mathbf{do} \\
 \quad \mathbf{if}\  odd(n) \ \mathbf{then}\  n:=3n+1 \  \mathbf{else}\ n:=n/2\  \mathbf{fi} \\
 \mathbf{od} \end{array}\right\} </math> <br />

@@ Linia 11: / Linia 11: @@
 <math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l}  \mathbf{while}\ n \neq 0 \ \mathbf{do} \\
 \quad \mathbf{if}\  odd(n) \ \mathbf{then}\  n:=3n+1 \  \mathbf{else}\ n:=n/2\  \mathbf{fi} \\
-\mathbf{od} \end{array}\right\} \color{black}</math> <br />
+\mathbf{od} \end{array}\right\} </math> <br />
 Note,  the truthfulness of the above sentence implies the truthfulness of Collatz's thesis, as it was formulated before World War II, in 1937.. <br />
 But then there were no computers or computer programming languages<br />

@@ Linia 7: / Linia 7: @@
 An abridged version of the proof is  [http://lem12.uksw.edu.pl/images/9/9f/MainPointsProofCollatz.pdf  here].<br />
 ==ntroduction==
-Rozpatrzmy zdanie<br/>
+Consider the following sentence<br/>
-dla każdej liczby naturalnej <math>n</math>, poniższy program <math>Cl </math> ma obliczenie sończone<br/>
+for every natural number  <math>n</math>, the program  <math>Cl </math> has the finite computation<br/>
 <math>\color{blue}\qquad Cl:\,\left\{\begin{array}{l}  \mathbf{while}\ n \neq 0 \ \mathbf{do} \\
 \quad \mathbf{if}\  nieparzyste(n) \ \mathbf{then}\  n:=3n+1 \  \mathbf{else}\ n:=n/2\  \mathbf{fi} \\
 \mathbf{od} \end{array}\right\} </math> <br />
-Zaczynamy od uwagi, że prawdziwośc powyższego zdania pociaga za sobą prawdzowość tezy Collatza, tak jak ona została sformułowana przed II wojną światową. <br />
+Note,  the truthfulness of the above sentence implies the truthfulness of Collatz's thesis, as it was formulated before World War II, in 1937.. <br />
-Ale w  r.1937  nie istniały komputery ani języki programowania<br />
+But then there were no computers or computer programming languages<br />
-.Z drugiej strony istniała i była już mocno rozwinięta teoria algorytmów. Teorię funkcji rekurencyjnych rozwijanow w Getyndze (David Hilbert i jego uczniowie), Budapeszcie (Rozsza Pterer, Laszlo Kalmar), ...<br />
+On the other hand, there was and was already a highly developed theory of algorithms. The theory of recursive functions was developed in Göttingen (David Hilbert and his students), Budapest (Rozsza Pterer, Laszlo Kalmar), ... <br />
-W Londynie Alan Turing stworzył abstrakcyjną maszynę Turinga.<br />
+In Princeton, Alonzo Church published papers on <math>\lambda</math>-calculus.
-W Moskwie Kołmogorow i w Kazaniu Malcew badali pojęcie funkcji obliczalnej. <br />
+In London, Alan Turing created the abstract Turing machine. <br />
 <br />
-W Warszawie Alfred Tarski wraz z uczniami Mojżeszem Presburgerem, Stanisławem Jaskowskim uzyskali ważne wyniki dotycące teorii dodawania liczb naturalnych.
+In Warsaw, Alfred Tarski and his students Mojżesz Presburger and Stanisław Jaskowski obtained important results on the theory of adding natural numbers.
 ==Remark made in  2004==

← poprzednia wersja		Wersja z 09:54, 28 wrz 2022
Linia 1:		Linia 1:
−	Collatz conjecture is valid in the standard structure of natural numbers, therefore it is Collatz theorem .<br />	+	Collatz conjecture is valid in the standard structure of natural numbers, therefore it is <big>Collatz theorem</big> .<br />

	Finally!<br />		Finally!<br />

@@ Linia 60: / Linia 60: @@
 [[Plik:MonStandardModel.png|center|thumb|600px|Nonstandard model of Presburger arithmetic]]
 The universe of the model is a subset of the set of complex numbers <math>a+\imath b</math>
-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,
+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>.
-with usual addition of complex numbers.
+Addition is defined as  usual addition of complex numbers.
 ==Algorithmic theory of natural numbers ==