= Hartogs numbers under choice
{c}
{title2=$h(X)=|X|^+$}
For infinite <sets> in <ZFC>, the <Hartogs number> is the successor of the input cardinal. Every smaller ordinal injects into the input, whereas that successor does not. An infinite <successor cardinal> $\kappa^+$ is regular: a cofinal family of at most $\kappa$ ordinals below it would have union of cardinal at most $\kappa\cdot\kappa=\kappa$, a contradiction. Finite inputs give finite successor cardinals, so the formulation in terms of alephs requires an infinite input. The general <Hartogs theorem> itself does not require <choice>.
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