Hartogs number

• Hartogs' number

After German-Jewish mathematician Friedrich Hartogs (1874–1943).

Hartogs number (plural Hartogs numbers)

1. (set theory) For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X.
   • 1973 [North-Holland], Thomas J. Jech, The Axiom of Choice, 2013, Dover, page 160,

     Let ${\displaystyle {\mathfrak {p}}}$ be an infinite cardinal, ${\displaystyle \vert X\vert ={\mathfrak {p}}}$ and let ${\displaystyle \aleph =\aleph ({\mathfrak {p}})}$ be the Hartogs number of ${\displaystyle {\mathfrak {p}}}$.

   • 1995, The Bulletin of Symbolic Logic, Volume 1, Association for Symbolic Logic, page 139,

     If the Power Set Axiom is replaced by "${\displaystyle \aleph (x)}$ is bound for every x" where

     ${\displaystyle \aleph (x)=\{a\vert \exists f(f}$ is one-to-one function from ${\displaystyle a}$ into ${\displaystyle x)\}}$,

     then the theory is denoted by ZFH (H stands for Hartogs' Number).

   • 2014, Barnaby Sheppard, The Logic of Infinity, Cambridge University Press, page 341:

     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.

• 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.

• equipotent
• Hartogs function