File:Peano-9.pdf

Summary

Description
English: Nonstandard-model of the Peano axioms, except for axiom 9 (induction). The horizontal vectors at the bottom denote the standard part, i.e. zero (leftmost, shown in glaring red) and its infinitely many -direct and indirect- successors. However, there is an additial nonstandard ("junk") part, viz. the cycle at the top, consisting of 48 members, none of which are reachable from zero via successor application. — See File:Domino effect visualizing exclusion of junk term by induction axiom.jpg for a 3d-graphic of the corrersponding domino effect. — See File:Peano-7.pdf and File:Peano-8.pdf for a nonstandard model violating just Peano axiom 7 and 8, respectively.
Date
Source Own work
Author Jochen Burghardt
PDF development
InfoField
LaTeX source code
\documentclass[12pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=300mm,paperheight=115mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{300mm}
\setlength{\textheight}{115mm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\setlength{\unitlength}{0.1mm}
\pagestyle{empty}

\definecolor{cMemZ}     {rgb}{0.90,0.00,0.00}   % zero

\definecolor{cMem0}     {rgb}{0.50,0.00,0.00}   % member
\definecolor{cMem1}     {rgb}{0.55,0.10,0.10}   % member
\definecolor{cMem2}     {rgb}{0.60,0.20,0.20}   % member
\definecolor{cMem3}     {rgb}{0.65,0.30,0.30}   % member
\definecolor{cMem4}     {rgb}{0.70,0.40,0.40}   % member
\definecolor{cMem5}     {rgb}{0.75,0.50,0.50}   % member
\definecolor{cMem6}     {rgb}{0.80,0.60,0.60}   % member
\definecolor{cMem7}     {rgb}{0.85,0.70,0.70}   % member
\definecolor{cMem8}     {rgb}{0.90,0.80,0.80}   % member
\definecolor{cMem9}     {rgb}{0.95,0.90,0.90}   % member (... fade out into infinity)

\definecolor{cMap0}     {rgb}{0.00,0.00,0.00}   % map arrow
\definecolor{cMap1}     {rgb}{0.10,0.10,0.10}   % map arrow
\definecolor{cMap2}     {rgb}{0.20,0.20,0.20}   % map arrow
\definecolor{cMap3}     {rgb}{0.30,0.30,0.30}   % map arrow
\definecolor{cMap4}     {rgb}{0.40,0.40,0.40}   % map arrow
\definecolor{cMap5}     {rgb}{0.50,0.50,0.50}   % map arrow
\definecolor{cMap6}     {rgb}{0.60,0.60,0.60}   % map arrow
\definecolor{cMap7}     {rgb}{0.70,0.70,0.70}   % map arrow
\definecolor{cMap8}     {rgb}{0.80,0.80,0.80}   % map arrow
\definecolor{cMap9}     {rgb}{0.90,0.90,0.90}   % map arrow  (... fade out into infinity)

\begin{document}
\begin{picture}(2950,1100)

% standard part map arrows:
\textcolor{cMap0}{\multiput(50.000,50.000)(100.000,0.000){10}{\vector(1,0){100.000}}}%
% fading out...
\textcolor{cMap0}{\put(1050.000,50.000){\vector(1,0){100.000}}}%
\textcolor{cMap0}{\put(1150.000,50.000){\vector(1,0){96.000}}}%
\textcolor{cMap0}{\put(1246.000,50.000){\vector(1,0){92.160}}}%
\textcolor{cMap0}{\put(1338.160,50.000){\vector(1,0){88.474}}}%
\textcolor{cMap0}{\put(1426.634,50.000){\vector(1,0){84.935}}}%
\textcolor{cMap0}{\put(1511.568,50.000){\vector(1,0){81.537}}}%
\textcolor{cMap0}{\put(1593.106,50.000){\vector(1,0){78.276}}}%
\textcolor{cMap0}{\put(1671.381,50.000){\vector(1,0){75.145}}}%
\textcolor{cMap0}{\put(1746.526,50.000){\vector(1,0){72.139}}}%
\textcolor{cMap0}{\put(1818.665,50.000){\vector(1,0){69.253}}}%
\textcolor{cMap0}{\put(1887.918,50.000){\vector(1,0){66.483}}}%
\textcolor{cMap0}{\put(1954.402,50.000){\vector(1,0){63.824}}}%
\textcolor{cMap0}{\put(2018.226,50.000){\vector(1,0){61.271}}}%
\textcolor{cMap0}{\put(2079.497,50.000){\vector(1,0){58.820}}}%
\textcolor{cMap0}{\put(2138.317,50.000){\vector(1,0){56.467}}}%
\textcolor{cMap0}{\put(2194.784,50.000){\vector(1,0){54.209}}}%
\textcolor{cMap0}{\put(2248.993,50.000){\vector(1,0){52.040}}}%
\textcolor{cMap0}{\put(2301.033,50.000){\vector(1,0){49.959}}}%
\textcolor{cMap0}{\put(2350.992,50.000){\vector(1,0){47.960}}}%
\textcolor{cMap0}{\put(2398.952,50.000){\vector(1,0){46.042}}}%
\textcolor{cMap0}{\put(2444.994,50.000){\vector(1,0){44.200}}}%
\textcolor{cMap1}{\put(2489.194,50.000){\vector(1,0){42.432}}}%
\textcolor{cMap2}{\put(2531.626,50.000){\vector(1,0){40.735}}}%
\textcolor{cMap3}{\put(2572.361,50.000){\vector(1,0){39.106}}}%
\textcolor{cMap4}{\put(2611.467,50.000){\vector(1,0){37.541}}}%
\textcolor{cMap5}{\put(2649.008,50.000){\vector(1,0){36.040}}}%
\textcolor{cMap6}{\put(2685.048,50.000){\vector(1,0){34.598}}}%
\textcolor{cMap7}{\put(2719.646,50.000){\vector(1,0){33.214}}}%
\textcolor{cMap8}{\put(2752.860,50.000){\vector(1,0){31.886}}}%
\textcolor{cMap9}{\put(2784.746,50.000){\vector(1,0){30.610}}}%

% standard part members:
\textcolor{cMemZ}{\put(50.000,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\multiput(150.000,50.000)(100.000,0.000){9}{\circle*{10.000}}}%
% fading out ...
\textcolor{cMem0}{\put(1050.000,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1150.000,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1246.000,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1338.160,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1426.634,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1511.568,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1593.106,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1671.381,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1746.526,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1818.665,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1887.918,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1954.402,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2018.226,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2079.497,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2138.317,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2194.784,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2248.993,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2301.033,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2350.992,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2398.952,50.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(2444.994,50.000){\circle*{10.000}}}%
\textcolor{cMem1}{\put(2489.194,50.000){\circle*{9.000}}}%
\textcolor{cMem2}{\put(2531.626,50.000){\circle*{9.000}}}%
\textcolor{cMem3}{\put(2572.361,50.000){\circle*{8.000}}}%
\textcolor{cMem4}{\put(2611.467,50.000){\circle*{8.000}}}%
\textcolor{cMem5}{\put(2649.008,50.000){\circle*{7.000}}}%
\textcolor{cMem6}{\put(2685.048,50.000){\circle*{7.000}}}%
\textcolor{cMem7}{\put(2719.646,50.000){\circle*{6.000}}}%
\textcolor{cMem8}{\put(2752.860,50.000){\circle*{6.000}}}%
\textcolor{cMem9}{\put(2784.746,50.000){\circle*{1.000}}}%
\textcolor{cMem9}{\put(2815.356,50.000){\circle*{1.000}}}%
\textcolor{cMem9}{\put(2844.742,50.000){\circle*{1.000}}}%
\textcolor{cMem9}{\put(2872.952,50.000){\circle*{1.000}}}%
\textcolor{cMem9}{\put(2900.034,50.000){\circle*{1.000}}}%

% nonstandard part map arrows
\textcolor{cMap0}{\put(1520.000,212.000){\vector(1,0){48.000}}}% +12,+0
\textcolor{cMap0}{\put(1568.000,212.000){\vector(4,1){48.000}}}% +12,+3
\textcolor{cMap0}{\put(1616.000,224.000){\vector(3,1){48.000}}}% +12,+4
\textcolor{cMap0}{\put(1664.000,240.000){\vector(2,1){48.000}}}% +12,+6
\textcolor{cMap0}{\put(1712.000,264.000){\vector(3,2){48.000}}}% +12,+8
\textcolor{cMap0}{\put(1760.000,296.000){\vector(4,3){48.000}}}% +12,+9
\textcolor{cMap0}{\put(1808.000,332.000){\vector(1,1){48.000}}}% +12,+12
\textcolor{cMap0}{\put(1856.000,380.000){\vector(3,4){36.000}}}% +9,+12
\textcolor{cMap0}{\put(1892.000,428.000){\vector(2,3){32.000}}}% +8,+12
\textcolor{cMap0}{\put(1924.000,476.000){\vector(1,2){24.000}}}% +6,+12
\textcolor{cMap0}{\put(1948.000,524.000){\vector(1,3){16.000}}}% +4,+12
\textcolor{cMap0}{\put(1964.000,572.000){\vector(1,4){12.000}}}% +3,+12
\textcolor{cMap0}{\put(1976.000,620.000){\vector(0,1){48.000}}}% +0,+12
\textcolor{cMap0}{\put(1976.000,668.000){\vector(-1,4){12.000}}}% -3,+12
\textcolor{cMap0}{\put(1964.000,716.000){\vector(-1,3){16.000}}}% -4,+12
\textcolor{cMap0}{\put(1948.000,764.000){\vector(-1,2){24.000}}}% -6,+12
\textcolor{cMap0}{\put(1924.000,812.000){\vector(-2,3){32.000}}}% -8,+12
\textcolor{cMap0}{\put(1892.000,860.000){\vector(-3,4){36.000}}}% -9,+12
\textcolor{cMap0}{\put(1856.000,908.000){\vector(-1,1){48.000}}}% -12,+12
\textcolor{cMap0}{\put(1808.000,956.000){\vector(-4,3){48.000}}}% -12,+9
\textcolor{cMap0}{\put(1760.000,992.000){\vector(-3,2){48.000}}}% -12,+8
\textcolor{cMap0}{\put(1712.000,1024.000){\vector(-2,1){48.000}}}% -12,+6
\textcolor{cMap0}{\put(1664.000,1048.000){\vector(-3,1){48.000}}}% -12,+4
\textcolor{cMap0}{\put(1616.000,1064.000){\vector(-4,1){48.000}}}% -12,+3
\textcolor{cMap0}{\put(1568.000,1076.000){\vector(-1,0){48.000}}}% -12,+0
\textcolor{cMap0}{\put(1520.000,1076.000){\vector(-4,-1){48.000}}}% -12,-3
\textcolor{cMap0}{\put(1472.000,1064.000){\vector(-3,-1){48.000}}}% -12,-4
\textcolor{cMap0}{\put(1424.000,1048.000){\vector(-2,-1){48.000}}}% -12,-6
\textcolor{cMap0}{\put(1376.000,1024.000){\vector(-3,-2){48.000}}}% -12,-8
\textcolor{cMap0}{\put(1328.000,992.000){\vector(-4,-3){48.000}}}% -12,-9
\textcolor{cMap0}{\put(1280.000,956.000){\vector(-1,-1){48.000}}}% -12,-12
\textcolor{cMap0}{\put(1232.000,908.000){\vector(-3,-4){36.000}}}% -9,-12
\textcolor{cMap0}{\put(1196.000,860.000){\vector(-2,-3){32.000}}}% -8,-12
\textcolor{cMap0}{\put(1164.000,812.000){\vector(-1,-2){24.000}}}% -6,-12
\textcolor{cMap0}{\put(1140.000,764.000){\vector(-1,-3){16.000}}}% -4,-12
\textcolor{cMap0}{\put(1124.000,716.000){\vector(-1,-4){12.000}}}% -3,-12
\textcolor{cMap0}{\put(1112.000,668.000){\vector(0,-1){48.000}}}% +0,-12
\textcolor{cMap0}{\put(1112.000,620.000){\vector(1,-4){12.000}}}% +3,-12
\textcolor{cMap0}{\put(1124.000,572.000){\vector(1,-3){16.000}}}% +4,-12
\textcolor{cMap0}{\put(1140.000,524.000){\vector(1,-2){24.000}}}% +6,-12
\textcolor{cMap0}{\put(1164.000,476.000){\vector(2,-3){32.000}}}% +8,-12
\textcolor{cMap0}{\put(1196.000,428.000){\vector(3,-4){36.000}}}% +9,-12
\textcolor{cMap0}{\put(1232.000,380.000){\vector(1,-1){48.000}}}% +12,-12
\textcolor{cMap0}{\put(1280.000,332.000){\vector(4,-3){48.000}}}% +12,-9
\textcolor{cMap0}{\put(1328.000,296.000){\vector(3,-2){48.000}}}% +12,-8
\textcolor{cMap0}{\put(1376.000,264.000){\vector(2,-1){48.000}}}% +12,-6
\textcolor{cMap0}{\put(1424.000,240.000){\vector(3,-1){48.000}}}% +12,-4
\textcolor{cMap0}{\put(1472.000,224.000){\vector(4,-1){48.000}}}% +12,-3
% nonstandard part members
\textcolor{cMem0}{\put(1520.000,212.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1568.000,212.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1616.000,224.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1664.000,240.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1712.000,264.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1760.000,296.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1808.000,332.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1856.000,380.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1892.000,428.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1924.000,476.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1948.000,524.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1964.000,572.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1976.000,620.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1976.000,668.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1964.000,716.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1948.000,764.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1924.000,812.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1892.000,860.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1856.000,908.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1808.000,956.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1760.000,992.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1712.000,1024.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1664.000,1048.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1616.000,1064.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1568.000,1076.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1520.000,1076.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1472.000,1064.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1424.000,1048.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1376.000,1024.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1328.000,992.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1280.000,956.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1232.000,908.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1196.000,860.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1164.000,812.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1140.000,764.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1124.000,716.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1112.000,668.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1112.000,620.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1124.000,572.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1140.000,524.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1164.000,476.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1196.000,428.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1232.000,380.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1280.000,332.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1328.000,296.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1376.000,264.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1424.000,240.000){\circle*{10.000}}}%
\textcolor{cMem0}{\put(1472.000,224.000){\circle*{10.000}}}%
\end{picture}
\end{document}

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
Category:CC-BY-SA-4.0#Peano-9.pdf
Category:Self-published work Category:Induction (mathematics) Category:Files by User:Jochen Burghardt Category:Images with LaTeX source code
Category:CC-BY-SA-4.0 Category:Files by User:Jochen Burghardt Category:Images with LaTeX source code Category:Induction (mathematics) Category:PNG created with LaTeX Category:Self-published work