Hartogs number

Etymology
After German-Jewish mathematician (1874–1943).

Noun

 * 1)  For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X.
 * 2) * 1973 [North-Holland],, The Axiom of Choice, 2013, Dover, page 160,
 * Let $$\mathfrak{p}$$ be an infinite cardinal, $$\vert X\vert = \mathfrak{p}$$ and let $$\aleph = \aleph(\mathfrak{p})$$ be the Hartogs number of $$\mathfrak{p}$$.
 * 1) * 1995, The Bulletin of Symbolic Logic, Volume 1,, page 139,
 * If the Power Set Axiom is replaced by "$$\aleph(x)$$ is bound for every x" where
 * $$\aleph(x) = \{a \vert \exists f(f$$ is one-to-one function from $$a$$ into $$x)\}$$,
 * then the theory is denoted by ZFH (H stands for Hartogs' Number).

Usage notes

 * The Hartogs number is a cardinal number representing the size of the ordinal number α (regarded as a set).
 * The definition is worded such that X does not need to have a well-order.