# A question about proving that there is no greatest cardinal

I would like to know if it is possible to prove that there is no greatest cardinal without axiom of choice

• – Winther Oct 28 '16 at 13:32

This is just Hartogs theorem, stating that if $X$ is any set, then there is an ordinal which admits no injections into $X$.
If $\alpha$ is any ordinal, then $\alpha$ can be mapped into $\mathcal P(\alpha)$, which has a larger cardinality, so by Hartogs theorem there is an ordinal of cardinality larger than $\alpha$. Namely, if $\alpha$ was a cardinal, then there has to be a larger cardinal than $\alpha$.