Hartogs number

Summary

In mathematics, specifically in axiomatic set theory, the Hartogs number of a set $X$ is the least ordinal number $\alpha$ such that there is no injection 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 of a set $A$). The existence of the Hartogs number of any $X$ can be proved in Zermelo–Fraenkel set theory (ZF) without relying on the axiom of choice. The map taking $X$ to $\alpha$ is sometimes called Hartogs's function.