Cantor’s paradox

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 $|𝒫(\alpha )|=|\alpha |$. (Here $𝒫(\alpha )$ denotes the power set of $\alpha$.) Suppose $f:\alpha \to 𝒫(\alpha )$ is a bijection proving their equicardinality. Then $X=\{\beta \in \alpha \mid \beta \notin f(\beta )\}$ 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.