Hartogs number

In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X. (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.) The map taking X to α is sometimes called Hartogs's function. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well-orderable sets.

The existence of the Hartogs number was proved by Friedrich Hartogs in 1915, using Zermelo–Fraenkel set theory alone (that is, without using the axiom of choice).