The **Hartog number** of a set $X$ is the least [ordinal](Ordinal.md "Ordinal") which cannot be mapped injectively into $X$. For well-ordered sets $X$ the Hartog number is exactly $\|X\|^+$, the [](Cardinal.md#Successor_cardinals) of $\|X\|$. When assuming the negation of the <a href="Axiom_of_Choice" class="mw-redirect" title="Axiom of Choice">axiom of choice</a> some sets cannot be well-ordered, and the Hartog number measures how well-ordered they can be. ## Properties - $X$ 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. This article is a stub. Please help us to improve Cantor's Attic by adding information.