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The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely: where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal.

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rdfs:label	始順序数 (ja) Von Neumann cardinal assignment (en) 冯·诺伊曼基数指派 (zh)
rdfs:comment	始順序数（しじゅんじょすう、英: initial ordinal）または始数（しすう、initial）とは、任意の順序数βに対してβ<α⇒card(β)濃度を表す) 。特に、濃度mの始順序数は、この定義により「濃度mの順序数のうち最小の順序数」となることからこのように呼ばれている (むしろこちらを定義とすることもある) 。また文献によっては、単にもっとも初めの順序数たる0を指すこともある。これは0が後続順序数、極限順序数のいずれでもないことを明示するが、形式的な分類名以上の意味はなく、つまらない。そのため、本記事では主に前者の意味での解説を行う。 (ja) 冯·诺伊曼基数指派是使用序数的基数指派。对于良序集合 U，我们定义它的基数为等势(equinumerous)于 U 的最小序数。更加精确的，，當中： * 是单射 * 和都为真為真 * 是序数的类。这个序数也叫做这个基数的初始序数。使用替代公理，U 是良序的和序数的类是良序的的事实保证这样一个序数存在并且是唯一的。通过完全选择公理，所有集合都是可良序的，所以所有集合都有一个基数；我们使用从序数继承来的次序排序基数。容易发现这与通过的排序相符。这是基数的良序排序。 (zh) The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely: where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal. (en)
dcterms:subject	Ordinal numbers Cardinal numbers
Wikipage page ID	373174 (xsd:integer)
Wikipage revision ID	1102794288 (xsd:integer)
Link from a Wikipage to another Wikipage	John von Neumann Countable set Order type Class (set theory) Epsilon numbers (mathematics) Equinumerosity Equivalence class Ordinal arithmetic Well-order Aleph number Cardinal assignment Cardinal number Ordinal numbers Cardinal numbers Axiom of choice Axiom schema of replacement Natural number Ordinal number Veblen function Uncountable set
sameAs	Von Neumann cardinal assignment Von Neumann cardinal assignment Von Neumann cardinal assignment Von Neumann cardinal assignment Von Neumann cardinal assignment Von Neumann cardinal assignment
dbp:wikiPageUsesTemplate	dbt:Refimprove dbt:Mathematical_logic
has abstract	The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely: where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal. That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤c. This is a well-ordering of cardinal numbers. (en) 始順序数（しじゅんじょすう、英: initial ordinal）または始数（しすう、initial）とは、任意の順序数βに対してβ<α⇒card(β)濃度を表す) 。特に、濃度mの始順序数は、この定義により「濃度mの順序数のうち最小の順序数」となることからこのように呼ばれている (むしろこちらを定義とすることもある) 。また文献によっては、単にもっとも初めの順序数たる0を指すこともある。これは0が後続順序数、極限順序数のいずれでもないことを明示するが、形式的な分類名以上の意味はなく、つまらない。そのため、本記事では主に前者の意味での解説を行う。 (ja) 冯·诺伊曼基数指派是使用序数的基数指派。对于良序集合 U，我们定义它的基数为等势(equinumerous)于 U 的最小序数。更加精确的，，當中： * 是单射 * 和都为真為真 * 是序数的类。这个序数也叫做这个基数的初始序数。使用替代公理，U 是良序的和序数的类是良序的的事实保证这样一个序数存在并且是唯一的。通过完全选择公理，所有集合都是可良序的，所以所有集合都有一个基数；我们使用从序数继承来的次序排序基数。容易发现这与通过的排序相符。这是基数的良序排序。 (zh)
prov:wasDerivedFrom	wikipedia-en:Von_Neumann_cardinal_assignment?oldid=1102794288&ns=0
page length (characters) of wiki page	4445 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Von_Neumann_cardinal_assignment
is Link from a Wikipage to another Wikipage of	John von Neumann Limit ordinal 0 Epsilon numbers (mathematics) Successor cardinal 1 Axiom of limitation of size Cardinal assignment Cardinal number Cardinality Axiom schema of replacement Natural number Ordinal number List of things named after John von Neumann Initial ordinal Von Neumann cardinal
is Wikipage redirect of	Initial ordinal Von Neumann cardinal
is foaf:primaryTopic of	wikipedia-en:Von_Neumann_cardinal_assignment

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