Hartogs number
التعريفات والمعاني
== English ==
=== Alternative forms ===
Hartogs' number
=== Etymology ===
After German-Jewish mathematician Friedrich Hartogs (1874–1943).
=== Noun ===
Hartogs number (plural Hartogs numbers)
(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
p
{\displaystyle {\mathfrak {p}}}
be an infinite cardinal,
|
X
|
=
p
{\displaystyle \vert X\vert ={\mathfrak {p}}}
and let
ℵ
=
ℵ
(
p
)
{\displaystyle \aleph =\aleph ({\mathfrak {p}})}
be the Hartogs number of
p
{\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 "
ℵ
(
x
)
{\displaystyle \aleph (x)}
is bound for every x" where
ℵ
(
x
)
=
{
a
|
∃
f
(
f
{\displaystyle \aleph (x)=\{a\vert \exists f(f}
is one-to-one function from
a
{\displaystyle a}
into
x
)
}
{\displaystyle x)\}}
,
then the theory is denoted by ZFH (H stands for Hartogs' Number).
==== 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 ====
=== See also ===
equipotent
Hartogs function