Hartogs number

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 ${\\displaystyle {\\mathfrak {p}}}$ 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.

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