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


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 $\alpha$.

It should also be noted that without the axiom of choice, the class of cardinals contains more than just ordinals. So appealing toe Cantor's theorem works just fine, as it ensures that there is no maximal cardinal.

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  • $\begingroup$ Thank you very much for your answer. $\endgroup$ – Daidalos Oct 28 '16 at 13:58

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