03-00-CantorsParadox.tex

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\pmtitle{Cantor's paradox}
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\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.
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