File:Peano-7.pdf

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English: Nonstandard-model of the Peano axioms, except for axiom 7 ("injectivity of successor"). Starting from zero (bottommost member, shown in glaring red), after 12 times applying the successor (i.e. following the arrow), a member is reached (shown in magenta) that is the successor of two distinct members (both shown in dark magenta); after 48 more applications of successor, the same element is reached again, this time from the other source. — See File:Peano-8.pdf and File:Peano-9.pdf for a nonstandard model violating just Peano axiom 8 and 9, respectively.
Date
Source Own work
Author Jochen Burghardt
Other versions File:Peano-7 svg.svg
LaTeX source code 
\documentclass[12pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=230mm,paperheight=300mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{230mm}
\setlength{\textheight}{300mm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\setlength{\unitlength}{0.1mm}
\pagestyle{empty}

\definecolor{cMemZ}     {rgb}{0.90,0.00,0.00}   % zero
\definecolor{cMemP}     {rgb}{0.50,0.00,0.50}   % two pred.s
\definecolor{cMemJ}     {rgb}{0.90,0.00,0.90}   % their common succ
\definecolor{cMem0}     {rgb}{0.50,0.00,0.00}   % member
\definecolor{cMap0}     {rgb}{0.00,0.00,0.00}   % map arrow

\begin{document}
\begin{picture}(2250,2950)
% initial part map arrows:
\textcolor{cMap0}{\put(680,50){\vector(4,1){180}}}% +180,+45
\textcolor{cMap0}{\put(860,95){\vector(3,1){180}}}% +180,+60
\textcolor{cMap0}{\put(1040,155){\vector(2,1){180}}}% +180,+90
\textcolor{cMap0}{\put(1220,245){\vector(3,2){180}}}% +180,+120
\textcolor{cMap0}{\put(1400,365){\vector(4,3){180}}}% +180,+135
\textcolor{cMap0}{\put(1580,500){\vector(1,1){180}}}% +180,+180
\textcolor{cMap0}{\put(1760,680){\vector(3,4){135}}}% +135,+180
\textcolor{cMap0}{\put(1895,860){\vector(2,3){120}}}% +120,+180
\textcolor{cMap0}{\put(2015,1040){\vector(1,2){90}}}% +90,+180
\textcolor{cMap0}{\put(2105,1220){\vector(1,3){60}}}% +60,+180
\textcolor{cMap0}{\put(2165,1400){\vector(1,4){45}}}% +45,+180
\textcolor{cMap0}{\put(2210,1580){\vector(0,1){180}}}% +0,+180
% cycle part map arrows:
\textcolor{cMap0}{\put(1070,740){\vector(1,0){120}}}% +120,+0
\textcolor{cMap0}{\put(1190,740){\vector(4,1){120}}}% +120,+30
\textcolor{cMap0}{\put(1310,770){\vector(3,1){120}}}% +120,+40
\textcolor{cMap0}{\put(1430,810){\vector(2,1){120}}}% +120,+60
\textcolor{cMap0}{\put(1550,870){\vector(3,2){120}}}% +120,+80
\textcolor{cMap0}{\put(1670,950){\vector(4,3){120}}}% +120,+90
\textcolor{cMap0}{\put(1790,1040){\vector(1,1){120}}}% +120,+120
\textcolor{cMap0}{\put(1910,1160){\vector(3,4){90}}}% +90,+120
\textcolor{cMap0}{\put(2000,1280){\vector(2,3){80}}}% +80,+120
\textcolor{cMap0}{\put(2080,1400){\vector(1,2){60}}}% +60,+120
\textcolor{cMap0}{\put(2140,1520){\vector(1,3){40}}}% +40,+120
\textcolor{cMap0}{\put(2180,1640){\vector(1,4){30}}}% +30,+120
\textcolor{cMap0}{\put(2210,1760){\vector(0,1){120}}}% +0,+120
\textcolor{cMap0}{\put(2210,1880){\vector(-1,4){30}}}% -30,+120
\textcolor{cMap0}{\put(2180,2000){\vector(-1,3){40}}}% -40,+120
\textcolor{cMap0}{\put(2140,2120){\vector(-1,2){60}}}% -60,+120
\textcolor{cMap0}{\put(2080,2240){\vector(-2,3){80}}}% -80,+120
\textcolor{cMap0}{\put(2000,2360){\vector(-3,4){90}}}% -90,+120
\textcolor{cMap0}{\put(1910,2480){\vector(-1,1){120}}}% -120,+120
\textcolor{cMap0}{\put(1790,2600){\vector(-4,3){120}}}% -120,+90
\textcolor{cMap0}{\put(1670,2690){\vector(-3,2){120}}}% -120,+80
\textcolor{cMap0}{\put(1550,2770){\vector(-2,1){120}}}% -120,+60
\textcolor{cMap0}{\put(1430,2830){\vector(-3,1){120}}}% -120,+40
\textcolor{cMap0}{\put(1310,2870){\vector(-4,1){120}}}% -120,+30
\textcolor{cMap0}{\put(1190,2900){\vector(-1,0){120}}}% -120,+0
\textcolor{cMap0}{\put(1070,2900){\vector(-4,-1){120}}}% -120,-30
\textcolor{cMap0}{\put(950,2870){\vector(-3,-1){120}}}% -120,-40
\textcolor{cMap0}{\put(830,2830){\vector(-2,-1){120}}}% -120,-60
\textcolor{cMap0}{\put(710,2770){\vector(-3,-2){120}}}% -120,-80
\textcolor{cMap0}{\put(590,2690){\vector(-4,-3){120}}}% -120,-90
\textcolor{cMap0}{\put(470,2600){\vector(-1,-1){120}}}% -120,-120
\textcolor{cMap0}{\put(350,2480){\vector(-3,-4){90}}}% -90,-120
\textcolor{cMap0}{\put(260,2360){\vector(-2,-3){80}}}% -80,-120
\textcolor{cMap0}{\put(180,2240){\vector(-1,-2){60}}}% -60,-120
\textcolor{cMap0}{\put(120,2120){\vector(-1,-3){40}}}% -40,-120
\textcolor{cMap0}{\put(80,2000){\vector(-1,-4){30}}}% -30,-120
\textcolor{cMap0}{\put(50,1880){\vector(0,-1){120}}}% +0,-120
\textcolor{cMap0}{\put(50,1760){\vector(1,-4){30}}}% +30,-120
\textcolor{cMap0}{\put(80,1640){\vector(1,-3){40}}}% +40,-120
\textcolor{cMap0}{\put(120,1520){\vector(1,-2){60}}}% +60,-120
\textcolor{cMap0}{\put(180,1400){\vector(2,-3){80}}}% +80,-120
\textcolor{cMap0}{\put(260,1280){\vector(3,-4){90}}}% +90,-120
\textcolor{cMap0}{\put(350,1160){\vector(1,-1){120}}}% +120,-120
\textcolor{cMap0}{\put(470,1040){\vector(4,-3){120}}}% +120,-90
\textcolor{cMap0}{\put(590,950){\vector(3,-2){120}}}% +120,-80
\textcolor{cMap0}{\put(710,870){\vector(2,-1){120}}}% +120,-60
\textcolor{cMap0}{\put(830,810){\vector(3,-1){120}}}% +120,-40
\textcolor{cMap0}{\put(950,770){\vector(4,-1){120}}}% +120,-30
% initial part members:
\textcolor{cMemZ}{\put(680,50){\circle*{10}}}%
\textcolor{cMem0}{\put(860,95){\circle*{10}}}%
\textcolor{cMem0}{\put(1040,155){\circle*{10}}}%
\textcolor{cMem0}{\put(1220,245){\circle*{10}}}%
\textcolor{cMem0}{\put(1400,365){\circle*{10}}}%
\textcolor{cMem0}{\put(1580,500){\circle*{10}}}%
\textcolor{cMem0}{\put(1760,680){\circle*{10}}}%
\textcolor{cMem0}{\put(1895,860){\circle*{10}}}%
\textcolor{cMem0}{\put(2015,1040){\circle*{10}}}%
\textcolor{cMem0}{\put(2105,1220){\circle*{10}}}%
\textcolor{cMem0}{\put(2165,1400){\circle*{10}}}%
\textcolor{cMemP}{\put(2210,1580){\circle*{10}}}%
% cycle part members:
\textcolor{cMem0}{\put(1070,740){\circle*{10}}}%
\textcolor{cMem0}{\put(1190,740){\circle*{10}}}%
\textcolor{cMem0}{\put(1310,770){\circle*{10}}}%
\textcolor{cMem0}{\put(1430,810){\circle*{10}}}%
\textcolor{cMem0}{\put(1550,870){\circle*{10}}}%
\textcolor{cMem0}{\put(1670,950){\circle*{10}}}%
\textcolor{cMem0}{\put(1790,1040){\circle*{10}}}%
\textcolor{cMem0}{\put(1910,1160){\circle*{10}}}%
\textcolor{cMem0}{\put(2000,1280){\circle*{10}}}%
\textcolor{cMem0}{\put(2080,1400){\circle*{10}}}%
\textcolor{cMem0}{\put(2140,1520){\circle*{10}}}%
\textcolor{cMemP}{\put(2180,1640){\circle*{10}}}%
\textcolor{cMemJ}{\put(2210,1760){\circle*{10}}}%
\textcolor{cMem0}{\put(2210,1880){\circle*{10}}}%
\textcolor{cMem0}{\put(2180,2000){\circle*{10}}}%
\textcolor{cMem0}{\put(2140,2120){\circle*{10}}}%
\textcolor{cMem0}{\put(2080,2240){\circle*{10}}}%
\textcolor{cMem0}{\put(2000,2360){\circle*{10}}}%
\textcolor{cMem0}{\put(1910,2480){\circle*{10}}}%
\textcolor{cMem0}{\put(1790,2600){\circle*{10}}}%
\textcolor{cMem0}{\put(1670,2690){\circle*{10}}}%
\textcolor{cMem0}{\put(1550,2770){\circle*{10}}}%
\textcolor{cMem0}{\put(1430,2830){\circle*{10}}}%
\textcolor{cMem0}{\put(1310,2870){\circle*{10}}}%
\textcolor{cMem0}{\put(1190,2900){\circle*{10}}}%
\textcolor{cMem0}{\put(1070,2900){\circle*{10}}}%
\textcolor{cMem0}{\put(950,2870){\circle*{10}}}%
\textcolor{cMem0}{\put(830,2830){\circle*{10}}}%
\textcolor{cMem0}{\put(710,2770){\circle*{10}}}%
\textcolor{cMem0}{\put(590,2690){\circle*{10}}}%
\textcolor{cMem0}{\put(470,2600){\circle*{10}}}%
\textcolor{cMem0}{\put(350,2480){\circle*{10}}}%
\textcolor{cMem0}{\put(260,2360){\circle*{10}}}%
\textcolor{cMem0}{\put(180,2240){\circle*{10}}}%
\textcolor{cMem0}{\put(120,2120){\circle*{10}}}%
\textcolor{cMem0}{\put(80,2000){\circle*{10}}}%
\textcolor{cMem0}{\put(50,1880){\circle*{10}}}%
\textcolor{cMem0}{\put(50,1760){\circle*{10}}}%
\textcolor{cMem0}{\put(80,1640){\circle*{10}}}%
\textcolor{cMem0}{\put(120,1520){\circle*{10}}}%
\textcolor{cMem0}{\put(180,1400){\circle*{10}}}%
\textcolor{cMem0}{\put(260,1280){\circle*{10}}}%
\textcolor{cMem0}{\put(350,1160){\circle*{10}}}%
\textcolor{cMem0}{\put(470,1040){\circle*{10}}}%
\textcolor{cMem0}{\put(590,950){\circle*{10}}}%
\textcolor{cMem0}{\put(710,870){\circle*{10}}}%
\textcolor{cMem0}{\put(830,810){\circle*{10}}}%
\textcolor{cMem0}{\put(950,770){\circle*{10}}}%
\end{picture}
\end{document}

Licensing

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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.

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Date/TimeThumbnailDimensionsUserComment
current15:56, 21 February 2015Thumbnail for version as of 15:56, 21 February 20151,356 × 1,770 (12 KB)Jochen Burghardt (talk | contribs)User created page with UploadWizard

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