Hartogs number

English

Alternative forms

• Hartogs' number

Etymology

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

Noun

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

   • 1995, The Bulletin of Symbolic Logic, Volume 1, Association for Symbolic Logic, 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).

   • 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.

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.

Translations

cardinality of the least ordinal for which there is no injection into a given set

See also

• equipotent
• Hartogs function