File:OrderTypeExamples.pdf

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Description
English: Three different strict well-orders on the natural numbers, each with a distinct order type. They correspond to three different ordinal numbers, but all to the same cardinal number (viz. ).
The red order cannot be order-isomorphic to the green , since the latter has a greatest element (viz. 4), while the former has not. For the same reason, the blue and the green order cannot be isomorphic. The red and the blue cannot be order-isomorphic, since the former has one limit point (i.e. corresponding to a limit ordinal), while the latter has two.
Date
Source Own work
Author Jochen Burghardt
Other versions File:OrderTypeExamples.pdf - File:OrderTypeExamples svg.svg
LaTeX source code
\documentclass[12pt]{article}

\setlength{\unitlength}{1mm}

\usepackage[pdftex]{color}
\usepackage{amssymb}
\usepackage[paperwidth=215mm,paperheight=95mm]{geometry}

\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{215mm}
\setlength{\textheight}{95mm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}

\pagestyle{empty}

% colors
\definecolor{fN}        {rgb}{0.50,0.00,0.00}   % \omega
\definecolor{fNV}       {rgb}{0.00,0.50,0.00}   % \omega+5
\definecolor{fNN}       {rgb}{0.00,0.00,0.50}   %\omega+\omega

\renewcommand{\leq}{\leqslant}
\newcommand{\N}{I\!\!N}

\newcommand{\curY}{(curY undefined)}

\newcommand{\NUM}[2]{%
        \put(#2,\curY){\put(0,-2){\line(0,1){4}}}%
        \put(#2,\curY){\put(0,3){\makebox(0,0)[b]{$#1$}}}%
}

\newcommand{\Num}[2]{%
        \put(#2,\curY){\put(0,-2){\line(0,1){4}}}%
        %\put(#2,\curY){\put(0,3){\makebox(0,0)[b]{$#1$}}}%
}

\newcommand{\num}[2]{%
        \put(#2,\curY){\put(0,-2){\line(0,1){4}}}%
        %\put(#2,\curY){\put(0,3){\makebox(0,0)[b]{$#1$}}}%
}

\begin{document}
\sf
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        %\put(-5,0){\makebox(0,0){$+$}}
        %\put(205,90){\makebox(0,0){$+$}}
\color{fN}%
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\put(0,\curY){\line(1,0){100}}%
%!}texfractal -geom 0.000 5.000 0.950 100
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\num{125}{99.836}%
\num{126}{99.844}%
\num{127}{99.852}%
\num{128}{99.859}%
\num{129}{99.866}%
\Num{130}{99.873}%
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\num{132}{99.885}%
\num{133}{99.891}%
\num{134}{99.896}%
\num{135}{99.902}%
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\num{137}{99.911}%
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\num{142}{99.931}%
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\num{189}{99.994}%
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\num{200}{99.996}%
%
\put(0,73){\makebox(0,0)[l]{Usual order $(<)$ on $\N$}}%
\put(0,68){\makebox(0,0)[l]{Order type ~ $\omega$}}%

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%
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\NUM{  3}{119.262}%
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%
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        Define ~
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        ~ as  ~
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        ~ or ~
        $m \mathrel{\textcolor{fN}{<}} n \mathrel{\textcolor{fN}{\leq}} 4$
        ~ or ~
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}}%
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\num{390}{ 99.995}%
\num{392}{ 99.996}%
\num{394}{ 99.996}%
\num{396}{ 99.996}%
\num{398}{ 99.996}%
\num{400}{ 99.996}%
%
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\NUM{  9}{123.549}%
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\num{ 13}{131.491}%
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\NUM{ 21}{145.126}%
\num{ 23}{148.120}%
\num{ 25}{150.964}%
\num{ 27}{153.666}%
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\num{233}{204.739}%
\num{235}{204.752}%
\num{237}{204.765}%
\num{239}{204.777}%
\Num{241}{204.788}%
\num{243}{204.798}%
\num{245}{204.808}%
\num{247}{204.818}%
\num{249}{204.827}%
\num{251}{204.836}%
\num{253}{204.844}%
\num{255}{204.852}%
\num{257}{204.859}%
\num{259}{204.866}%
\Num{261}{204.873}%
\num{263}{204.879}%
\num{265}{204.885}%
\num{267}{204.891}%
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\num{271}{204.902}%
\num{273}{204.907}%
\num{275}{204.911}%
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\num{279}{204.920}%
\Num{281}{204.924}%
\num{283}{204.928}%
\num{285}{204.931}%
\num{287}{204.935}%
\num{289}{204.938}%
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\num{299}{204.952}%
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\num{329}{204.978}%
\num{331}{204.979}%
\num{333}{204.980}%
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\num{337}{204.982}%
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\num{343}{204.984}%
\num{345}{204.985}%
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\num{349}{204.987}%
\num{351}{204.987}%
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\num{363}{204.991}%
\num{365}{204.991}%
\num{367}{204.992}%
\num{369}{204.992}%
\num{371}{204.992}%
\num{373}{204.993}%
\num{375}{204.993}%
\num{377}{204.994}%
\num{379}{204.994}%
\Num{381}{204.994}%
\num{383}{204.994}%
\num{385}{204.995}%
\num{387}{204.995}%
\num{389}{204.995}%
\num{391}{204.995}%
\num{393}{204.996}%
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\num{397}{204.996}%
\num{399}{204.996}%
\num{401}{204.996}%
%
\put(0,13){\makebox(0,0)[l]{%
        Define ~
        $m \sqsubset n$
        ~ as  ~
        ( $m+n$ even
        ~ and ~
        $m \mathrel{\textcolor{fN}{<}} n$ )
        ~ or ~
        ( $m$ even
        ~ and ~
        $n$ odd )
}}%
\put(0,8){\makebox(0,0)[l]{Order type ~ $\omega+\omega$}}%

\end{picture}
\end{document}

Licensing

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