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.