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  "body": "\n## Summary\n\nIn 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**.\n",
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