# Hartog number

From Cantor's Attic

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.

## 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.

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