---
tag: 2SQ1WWQ7TK
kind: concept
name: Hartogs number
aliases: [hartogs-numbers, hartogs-function]
sources:
  - wikipedia: en:2977752@r1339480874
    locator: Hartogs number
    note: adopted
    mention: wikipedia://en/2977752
    sections: [Summary]
same_as:
  wikidata: Q1186007
  wikipedia: Hartogs number
inventory: [MS1AABYPY3]
---

## Summary

In mathematics, specifically in axiomatic set theory, the **Hartogs number** of a [[set|0BSJ7RQ9BW]] $X$ is the least [[ordinal number|0RDJK7EQYY]] $\alpha$ such that there is no [[injection|09NWETVTNY]] from $\alpha$ into $X$. In other words, $\alpha$ is the least ordinal such that $|\alpha| \not \le |X|$ (where $\lvert A\rvert$ denotes the [[cardinality|0G0ASKNA6Z]] of a set $A$). The existence of the Hartogs number of any $X$ can be proved in [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] (ZF) without relying on the [[axiom of choice|0HTGB13FE2]]. The map taking $X$ to $\alpha$ is sometimes called **Hartogs's function**.