File:Diagonal argument powerset.pdf
Summary
| Description |
English: An illustration of Cantor's diagonal argument for the existence of sets of arbitrary large cardinality (Cantor's theorem). The set T = {n∈ℕ: n∉f(n)} at the bottom cannot occur anywhere in the range of f:ℕ→ℙ(ℕ). The example mapping f is chosen similar to the example enumeration s in File:Diagonal argument 01.pdf. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:Diagonal argument powerset svg.svg |
LaTeX source Code |
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\documentclass{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=75mm,paperheight=65mm]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{75mm}
\setlength{\textheight}{65mm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0mm}
\definecolor{cPos} {rgb}{0.99,0.90,0.90}
\definecolor{cNeg} {rgb}{0.90,0.90,0.99}
\definecolor{cPOS} {rgb}{0.99,0.00,0.00}
\definecolor{cNEG} {rgb}{0.00,0.00,0.99}
\pagestyle{empty}
\newcommand{\0}[1]{\textcolor{cPos}{#1}}
\newcommand{\1}[1]{\textcolor{cNeg}{#1}}
\newcommand{\2}[1]{\textcolor{cPOS}{#1}}
\newcommand{\3}[1]{\textcolor{cNEG}{#1}}
\begin{document}
$\begin{array}{|*{15}{@{\;}c}|}
\hline
f(1) & = \{ & \0{1,} & & & & & & & & & & & & \} \\
f(2) & = \{ & 1, & \2{2,}& 3, & 4, & 5, & 6, & 7, & 8, & 9, & 10, & 11, & \ldots & \} \\
f(3) & = \{ & & 2, & \0{3,}& 4, & & 6, & & 8, & & 10, & & \ldots & \} \\
f(4) & = \{ & 1, & & 3, &\0{4,} & 5, & & 7, & & 9, & & 11, & \ldots & \} \\
f(5) & = \{ & 1, & 2, & & 4, &\0{5,} & 6, & 7, & & 9, & & 11, & \ldots & \} \\
f(6) & = \{ & & & 3, & 4, & & \2{6,}& 7, & & 9, & 10, & & \ldots & \} \\
f(7) & = \{ & 1, & & & & 5, & &\0{7,} & & 9, & & & \ldots & \} \\
f(8) & = \{ & & & 3, & 4, & & & 7, & \2{8,}& & & 11, & \ldots & \} \\
f(9) & = \{ & 1, & 2, & & & 5, & 6, & & & \2{9,}& 10, & & \ldots & \} \\
f(10) & = \{ & 1, & 2, & & 4, & 5, & 6, & & & 9, &\0{10,}& 11, & \ldots & \} \\
f(11) & = \{ & 1, & 2, & & 4, & & 6, & & & 9, & &\0{11},& \ldots & \} \\
\vdots & & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \\
\hline
\multicolumn{10}{c}{} \\
\hline
T & = \{ & \3{1,}& \1{2,}& \3{3,}& \3{4,}& \3{5,}& \1{6,}& \3{7,}& \1{8,}& \1{9,}& \3{10,}& \3{11,} & \ldots & \} \\
\hline
\end{array}$
\end{document}
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