Meaning of Hartogs number | Babel Free

Definitions

For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X.

Examples

“1973 [North-Holland], Thomas J. Jech, The Axiom of Choice, 2013, Dover, page 160, Let p be an infinite cardinal, |X|=p and let א=א(p) be the Hartogs number of p.”

“1995, The Bulletin of Symbolic Logic, Volume 1, Association for Symbolic Logic, page 139, If the Power Set Axiom is replaced by "א(x) is bound for every x" where א(x)={a|∃f(f is one-to-one function from a into x)}, then the theory is denoted by ZFH (H stands for Hartogs' Number).”

“Since the proof of Hartogs' Theorem does not appeal to the Axiom of Choice, the Hartogs number of a set X exists whether or not X has a well-ordering.”

CEFR level