File:Peano-7.pdf
Summary
| Description |
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}
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