File:Peano-8.pdf
Summary
| Description |
English: Nonstandard-model of the Peano axioms, except for axiom 8 ("zero not in range of successor"). Starting from zero (at the bottom, shown in glaring red), after 48 times applying the successor (i.e. following the arrow), zero is reached again. — See File:Peano-7.pdf and File:Peano-9.pdf for a nonstandard model violating just Peano axiom 7 and 9, respectively. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:Peano-8 svg.svg |
LaTeX source code |
|---|
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\usepackage[pdftex]{color}
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\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{230mm}
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\pagestyle{empty}
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\definecolor{cMem0} {rgb}{0.50,0.00,0.00} % member
\definecolor{cMap0} {rgb}{0.00,0.00,0.00} % map arrow
\begin{document}
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% map arrows
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\textcolor{cMap0}{\put(1170,30){\vector(4,1){120}}}% +12,+3
\textcolor{cMap0}{\put(1290,60){\vector(3,1){120}}}% +12,+4
\textcolor{cMap0}{\put(1410,100){\vector(2,1){120}}}% +12,+6
\textcolor{cMap0}{\put(1530,160){\vector(3,2){120}}}% +12,+8
\textcolor{cMap0}{\put(1650,240){\vector(4,3){120}}}% +12,+9
\textcolor{cMap0}{\put(1770,330){\vector(1,1){120}}}% +12,+12
\textcolor{cMap0}{\put(1890,450){\vector(3,4){90}}}% +9,+12
\textcolor{cMap0}{\put(1980,570){\vector(2,3){80}}}% +8,+12
\textcolor{cMap0}{\put(2060,690){\vector(1,2){60}}}% +6,+12
\textcolor{cMap0}{\put(2120,810){\vector(1,3){40}}}% +4,+12
\textcolor{cMap0}{\put(2160,930){\vector(1,4){30}}}% +3,+12
\textcolor{cMap0}{\put(2190,1050){\vector(0,1){120}}}% +0,+12
\textcolor{cMap0}{\put(2190,1170){\vector(-1,4){30}}}% -3,+12
\textcolor{cMap0}{\put(2160,1290){\vector(-1,3){40}}}% -4,+12
\textcolor{cMap0}{\put(2120,1410){\vector(-1,2){60}}}% -6,+12
\textcolor{cMap0}{\put(2060,1530){\vector(-2,3){80}}}% -8,+12
\textcolor{cMap0}{\put(1980,1650){\vector(-3,4){90}}}% -9,+12
\textcolor{cMap0}{\put(1890,1770){\vector(-1,1){120}}}% -12,+12
\textcolor{cMap0}{\put(1770,1890){\vector(-4,3){120}}}% -12,+9
\textcolor{cMap0}{\put(1650,1980){\vector(-3,2){120}}}% -12,+8
\textcolor{cMap0}{\put(1530,2060){\vector(-2,1){120}}}% -12,+6
\textcolor{cMap0}{\put(1410,2120){\vector(-3,1){120}}}% -12,+4
\textcolor{cMap0}{\put(1290,2160){\vector(-4,1){120}}}% -12,+3
\textcolor{cMap0}{\put(1170,2190){\vector(-1,0){120}}}% -12,+0
\textcolor{cMap0}{\put(1050,2190){\vector(-4,-1){120}}}% -12,-3
\textcolor{cMap0}{\put(930,2160){\vector(-3,-1){120}}}% -12,-4
\textcolor{cMap0}{\put(810,2120){\vector(-2,-1){120}}}% -12,-6
\textcolor{cMap0}{\put(690,2060){\vector(-3,-2){120}}}% -12,-8
\textcolor{cMap0}{\put(570,1980){\vector(-4,-3){120}}}% -12,-9
\textcolor{cMap0}{\put(450,1890){\vector(-1,-1){120}}}% -12,-12
\textcolor{cMap0}{\put(330,1770){\vector(-3,-4){90}}}% -9,-12
\textcolor{cMap0}{\put(240,1650){\vector(-2,-3){80}}}% -8,-12
\textcolor{cMap0}{\put(160,1530){\vector(-1,-2){60}}}% -6,-12
\textcolor{cMap0}{\put(100,1410){\vector(-1,-3){40}}}% -4,-12
\textcolor{cMap0}{\put(60,1290){\vector(-1,-4){30}}}% -3,-12
\textcolor{cMap0}{\put(30,1170){\vector(0,-1){120}}}% +0,-12
\textcolor{cMap0}{\put(30,1050){\vector(1,-4){30}}}% +3,-12
\textcolor{cMap0}{\put(60,930){\vector(1,-3){40}}}% +4,-12
\textcolor{cMap0}{\put(100,810){\vector(1,-2){60}}}% +6,-12
\textcolor{cMap0}{\put(160,690){\vector(2,-3){80}}}% +8,-12
\textcolor{cMap0}{\put(240,570){\vector(3,-4){90}}}% +9,-12
\textcolor{cMap0}{\put(330,450){\vector(1,-1){120}}}% +12,-12
\textcolor{cMap0}{\put(450,330){\vector(4,-3){120}}}% +12,-9
\textcolor{cMap0}{\put(570,240){\vector(3,-2){120}}}% +12,-8
\textcolor{cMap0}{\put(690,160){\vector(2,-1){120}}}% +12,-6
\textcolor{cMap0}{\put(810,100){\vector(3,-1){120}}}% +12,-4
\textcolor{cMap0}{\put(930,60){\vector(4,-1){120}}}% +12,-3
% members
\textcolor{cMem0}{\put(1050,30){\circle*{10}}}%
\textcolor{cMemZ}{\put(1170,30){\circle*{10}}}%
\textcolor{cMem0}{\put(1290,60){\circle*{10}}}%
\textcolor{cMem0}{\put(1410,100){\circle*{10}}}%
\textcolor{cMem0}{\put(1530,160){\circle*{10}}}%
\textcolor{cMem0}{\put(1650,240){\circle*{10}}}%
\textcolor{cMem0}{\put(1770,330){\circle*{10}}}%
\textcolor{cMem0}{\put(1890,450){\circle*{10}}}%
\textcolor{cMem0}{\put(1980,570){\circle*{10}}}%
\textcolor{cMem0}{\put(2060,690){\circle*{10}}}%
\textcolor{cMem0}{\put(2120,810){\circle*{10}}}%
\textcolor{cMem0}{\put(2160,930){\circle*{10}}}%
\textcolor{cMem0}{\put(2190,1050){\circle*{10}}}%
\textcolor{cMem0}{\put(2190,1170){\circle*{10}}}%
\textcolor{cMem0}{\put(2160,1290){\circle*{10}}}%
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\textcolor{cMem0}{\put(2060,1530){\circle*{10}}}%
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\textcolor{cMem0}{\put(1770,1890){\circle*{10}}}%
\textcolor{cMem0}{\put(1650,1980){\circle*{10}}}%
\textcolor{cMem0}{\put(1530,2060){\circle*{10}}}%
\textcolor{cMem0}{\put(1410,2120){\circle*{10}}}%
\textcolor{cMem0}{\put(1290,2160){\circle*{10}}}%
\textcolor{cMem0}{\put(1170,2190){\circle*{10}}}%
\textcolor{cMem0}{\put(1050,2190){\circle*{10}}}%
\textcolor{cMem0}{\put(930,2160){\circle*{10}}}%
\textcolor{cMem0}{\put(810,2120){\circle*{10}}}%
\textcolor{cMem0}{\put(690,2060){\circle*{10}}}%
\textcolor{cMem0}{\put(570,1980){\circle*{10}}}%
\textcolor{cMem0}{\put(450,1890){\circle*{10}}}%
\textcolor{cMem0}{\put(330,1770){\circle*{10}}}%
\textcolor{cMem0}{\put(240,1650){\circle*{10}}}%
\textcolor{cMem0}{\put(160,1530){\circle*{10}}}%
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\textcolor{cMem0}{\put(100,810){\circle*{10}}}%
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\textcolor{cMem0}{\put(240,570){\circle*{10}}}%
\textcolor{cMem0}{\put(330,450){\circle*{10}}}%
\textcolor{cMem0}{\put(450,330){\circle*{10}}}%
\textcolor{cMem0}{\put(570,240){\circle*{10}}}%
\textcolor{cMem0}{\put(690,160){\circle*{10}}}%
\textcolor{cMem0}{\put(810,100){\circle*{10}}}%
\textcolor{cMem0}{\put(930,60){\circle*{10}}}%
\end{picture}
\end{document}
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