130.225.98.243: /* Cofinality of cardinals */ AC implies that cardinals are ordinals, so any nonempty subset has a least element. No need for infimum.

2014-11-13T14:34:08Z

Cofinality of cardinals: AC implies that cardinals are ordinals, so any nonempty subset has a least element. No need for infimum.

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	In [[mathematics]], especially in [[order theory]], the '''cofinality''' cf(''A'') of a [[partially ordered set]] ''A'' is the least of the [[cardinality\|cardinalities]] of the [[cofinal (mathematics)\|cofinal]] subsets of ''A''.

	This definition of cofinality relies on the [[axiom of choice]], as it uses the fact that every non-empty set of [[cardinal number]]s has a least member. The cofinality of a partially ordered set ''A'' can alternatively be defined as the least [[ordinal number\|ordinal]] ''x'' such that there is a function from ~~''x'' to ''A'' with cofinal [[Image (mathematics)\|image]]~~. ~~This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.~~

	~~Cofinality can be similarly defined for a [[directed set]] and is used to generalize the notion of a [[subsequence]] in a [[Net (mathematics)\|net]].~~

	~~== Examples ==~~
	* The cofinality of a partially ordered set with [[greatest element]] is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subset).
	** In particular, the cofinality of any nonzero finite ordinal, or indeed any finite directed set, is 1, since such sets have a greatest element.
	* Every cofinal subset of a partially ordered set must contain all [[maximal element]]s of that set. Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.
	** In particular, let ''A'' be a set of size ''n'', and consider the set of subsets of ''A'' containing no more than ''m~~'' elements~~. ~~This~~ is ~~partially ordered under inclusion and the subsets with ''m'' elements are maximal~~. ~~Thus the cofinality of this poset is ''n'' [[Binomial coefficient\|choose]] ''m''.~~
	* A subset of the natural numbers '''N''' is cofinal in '''N''' if and only if it is infinite, and therefore the cofinality of &alefsym;<~~sub~~>~~0</sub> is &alefsym;<sub>0</sub>. Thus &alefsym;<sub>0</sub> is~~ a ~~[[regular cardinal]]~~.
	* The cofinality of the [[real number]]s with their usual ordering is &alefsym;<~~sub~~>~~0</sub>, since '''N''' is cofinal~~ in '''R'''. The usual ordering of '''R''' is not [[order isomorphic]] to ''c'', the [[cardinality of the continuum\|cardinality of the real numbers]], which has cofinality strictly greater than &alefsym;<sub>0</sub>. This demonstrates that the cofinality depends on the order; different orders on the same set may have different cofinality.

	~~==Properties==~~
	~~If ''A'' admits a [[total order\|totally ordered]] cofinal subset, then we can find a subset ''B'' which is well-ordered and cofinal~~ in ~~''A''~~. ~~Any subset of ''B'' is also well-ordered. If two cofinal subsets of ''B'' have minimal cardinality (i.e. their cardinality is the cofinality of ''B''), then they are order isomorphic to each other.~~

	~~== Cofinality of ordinals and other well-ordered sets ==~~
	The '''cofinality of an ordinal''' α is the smallest ordinal δ which is the [[order type]] of a [[cofinal subset]] of α. The cofinality of a set of ordinals or any other [[well-ordered set]] is the cofinality of the order type of that set.

	~~Thus for a [[limit ordinal]], there exists a δ-indexed strictly increasing sequence with limit α. For example, the cofinality of ω² is ω, because the sequence ω·''m'' (where '~~'m'' ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does ω<sub>ω</sub> or an uncountable cofinality.

	~~The cofinality of 0 is 0. The cofinality of any [[successor ordinal]] is 1. The cofinality of any limit ordinal is~~ at ~~least ω.~~

	~~== Regular and singular ordinals ==~~
	~~A '''regular ordinal''' is an ordinal which is equal to its cofinality. A '''singular ordinal''' is any ordinal which is not regular.~~

	~~Every regular ordinal is~~ the ~~[[initial ordinal]] of a cardinal~~. ~~Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular. Assuming the Axiom of choice,~~ <~~math~~>~~\omega_{\alpha+1}~~<~~/math~~> is regular for each α. In this case, the ordinals 0, 1, <math>\omega</math>, <math>\omega_1</math>, and <math>\omega_2</math> are regular, whereas 2, 3, <math>\omega_\omega</math>, and ω<sub>ω·2</sub> are initial ordinals which are not regular.

	~~The cofinality of any ordinal ''α'' is a regular ordinal, i.e. the cofinality of the cofinality of ''α'' is the same as the cofinality of ''α''. So the cofinality operation is [[idempotent]].~~

	~~==Cofinality of cardinals==~~
	~~If κ is an infinite cardinal number, then cf(κ) is the least cardinal such that there is an [[bounded (set theory)\|unbounded]] function from it to κ~~; ~~and cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ; more precisely~~

	~~:<math>\mathrm{cf}(\kappa) = \inf \left\{ \mathrm{card}(I)\ \|\ \kappa = \sum_{i \in I} \lambda_i\ \mathrm{and}\ (\forall i)(\lambda_i < \kappa)\right\}</math>~~

	~~That the set above is nonempty comes from the fact that~~

	~~:<math>\kappa = \bigcup_{i \in \kappa} \{i\}</math>~~

	~~i.e. the~~ [~~[disjoint union]] of κ singleton sets. This implies immediately that cf(κ) ≤ κ.~~
	~~The cofinality of any totally ordered set is regular, so one has cf(κ) = cf(cf(κ)).~~

	~~Using [[König's theorem (set theory)\|König's theorem]], one can prove κ < κ<sup>cf(κ)</sup> and κ < cf(2<sup>κ</sup>) for any infinite cardinal κ.~~

	~~The last inequality implies that the cofinality of the cardinality of the continuum must be uncountable. On the other hand,~~

	:~~<math> \aleph_\omega = \bigcup_{n < \omega} \aleph_n </math>.~~

	~~the ordinal number ω being the first infinite ordinal, so that the cofinality of <math>\aleph_\omega<~~/~~math> is card(ω) = <math>\aleph_0<~~/~~math>~~. ~~(In particular, <math>\aleph_\omega</math> is singular~~.~~) Therefore,~~

	~~:<math>2^{\aleph_0}\neq\aleph_\omega.</math>~~

	~~(Compare to the [[continuum hypothesis]], which states <math>2^{\aleph_0}= \aleph_1<~~/~~math>~~.)

	~~Generalizing this argument, one can prove that for a limit ordinal δ~~

	~~:<math>\mathrm{cf} (\aleph_\delta) = \mathrm{cf} (\delta) </math>.~~

	~~==See also==~~
	*[[Initial ordinal]]

	~~==References==~~
	*Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
	*Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.

	~~[[Category:Order theory]]~~
	~~[[Category:Set theory]]~~
	~~[[Category:Ordinal numbers]]~~
	~~[[Category:Cardinal numbers]~~]

en>Legobot: Bot: Migrating langlinks to WP:Wikidata - d:q1283623

2013-02-20T16:33:37Z

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en>Xantharius: /* Cofinality of ordinals and other well-ordered sets */ Unified math styles as per WP:MOS

2012-05-07T18:54:51Z

Cofinality of ordinals and other well-ordered sets: Unified math styles as per WP:MOS

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Cofinality - Revision history

130.225.98.243: /* Cofinality of cardinals */ AC implies that cardinals are ordinals, so any nonempty subset has a least element. No need for infimum.

en>Legobot: Bot: Migrating langlinks to WP:Wikidata - d:q1283623

en>Xantharius: /* Cofinality of ordinals and other well-ordered sets */ Unified math styles as per WP:MOS