CantorsParadox 2002-09-29 15:23:13 2005-02-14 19:49:21 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} \emph{Cantor's paradox} demonstrates that there can be no largest cardinality. In particular, there must be an unlimited number of infinite cardinalities. For suppose that $\alpha$ were the largest cardinal. Then we would have $|\mathcal{P}(\alpha)|=|\alpha|$. (Here $\mathcal{P}(\alpha)$ denotes the power set of $\alpha$.) Suppose $f:\alpha\rightarrow\mathcal{P}(\alpha)$ is a bijection proving their equicardinality. Then $X=\{\beta\in\alpha\mid \beta\not\in f(\beta)\}$ is a subset of $\alpha$, and so there is some $\gamma\in\alpha$ such that $f(\gamma)=X$. But $\gamma\in X\leftrightarrow\gamma\notin X$, which is a paradox. The key part of the argument strongly resembles Russell's paradox, which is in some sense a generalization of this paradox. Besides allowing an unbounded number of cardinalities as ZF set theory does, this paradox could be avoided by a few other tricks, for instance by not allowing the construction of a power set or by adopting paraconsistent logic.