Hartogs number

English

Alternative forms

• Template:L

Etymology

After German-Jewish mathematician Template:W (1874–1943).

Noun

1. Template:Lb 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], Template:W, The Axiom of Choice, 2013, Dover, page 160,

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

   • 1995, The Bulletin of Symbolic Logic, Volume 1, Template:W, page 139,

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

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

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

   • Template:Quote-book

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.

See also

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