Hartog number

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The Hartog number of a set $X$ is the least ordinal which cannot be mapped injectively into $X$. For well-ordered sets $X$ the Hartog number is exactly $|X|^+$, the successor cardinal of $|X|$.

When assuming the negation of the axiom of choice some sets cannot be well-ordered, and the Hartog number measures how well-ordered they can be.


  • $X$ can be well ordered if and only if $|X|$ is comparable with its Hartog number.
  • There can be a surjection from $X$ onto its Hartog number.

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