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 were the largest cardinal. Then we would have . (Here denotes the power set of .) Suppose is a bijection proving their equicardinality. Then is a subset of , and so there is some such that . But , 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.