Editing Cofinality

{{Short description|Size of subsets in order theory}}
{{Distinguish|cofiniteness}}
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''. Formally,<ref>{{cite arXiv |last=Shelah |first=Saharon |date=26 November 2002 |title=Logical Dreams |eprint=math/0211398}}</ref>
:<math>\operatorname{cf}(A) = \inf \{|B| : B \subseteq A, (\forall x \in A) (\exists y \in B) (x \leq y)\}</math>

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 <math>A</math> be a set of size <math>n,</math> and consider the set of subsets of <math>A</math> containing no more than <math>m</math> elements.  This is partially ordered under inclusion and the subsets with <math>m</math> elements are maximal.  Thus the cofinality of this poset is <math>n</math> [[Binomial coefficient|choose]] <math>m.</math>
* A subset of the [[natural number]]s <math>\N</math> is cofinal in <math>\N</math> if and only if it is infinite, and therefore the cofinality of <math>\aleph_0</math> is <math>\aleph_0.</math> Thus <math>\aleph_0</math> is a [[regular cardinal]].
* The cofinality of the [[real number]]s with their usual ordering is <math>\aleph_0,</math> since <math>\N</math> is cofinal in <math>\R.</math>  The usual ordering of <math>\R</math> is not [[order isomorphic]] to <math>c,</math> the [[Cardinality of the continuum|cardinality of the real numbers]], which has cofinality strictly greater than <math>\aleph_0.</math>  This demonstrates that the cofinality depends on the order; different orders on the same set may have different cofinality.

==Properties==

If <math>A</math> admits a [[total order|totally ordered]] cofinal subset, then we can find a subset <math>B</math> that is well-ordered and cofinal in <math>A.</math> Any subset of <math>B</math> is also well-ordered. Two cofinal subsets of <math>B</math> with minimal cardinality (that is, their cardinality is the cofinality of <math>B</math>) need not be order isomorphic (for example if <math>B = \omega + \omega,</math> then both <math>\omega + \omega</math> and <math>\{\omega + n : n < \omega\}</math> viewed as subsets of <math>B</math> have the countable cardinality of the cofinality of <math>B</math> but are not order isomorphic).  But cofinal subsets of <math>B</math> with minimal order type will be order isomorphic.

==Cofinality of ordinals and other well-ordered sets==

The '''cofinality of an ordinal''' <math>\alpha</math> is the smallest ordinal <math>\delta</math> that is the [[order type]] of a [[cofinal subset]] of <math>\alpha.</math>  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]] <math>\alpha,</math> there exists a <math>\delta</math>-indexed strictly increasing sequence with limit <math>\alpha.</math>  For example, the cofinality of <math>\omega^2</math> is <math>\omega,</math> because the sequence <math>\omega \cdot m</math> (where <math>m</math> ranges over the natural numbers) tends to <math>\omega^2;</math> but, more generally, any countable limit ordinal has cofinality <math>\omega.</math>  An uncountable limit ordinal may have either cofinality <math>\omega</math> as does <math>\omega_\omega</math> or an uncountable cofinality.

The cofinality of 0 is 0. The cofinality of any [[successor ordinal]] is 1.  The cofinality of any nonzero limit ordinal is an infinite regular cardinal.

==Regular and singular ordinals==
{{Main|Regular cardinal}}

A '''regular ordinal''' is an ordinal that is equal to its cofinality. A '''singular ordinal''' is any ordinal that 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 <math>\alpha.</math>  In this case, the ordinals <math>0, 1, \omega, \omega_1,</math> and <math>\omega_2</math> are regular, whereas <math>2, 3, \omega_\omega,</math> and <math>\omega_{\omega \cdot 2}</math> are initial ordinals that are not regular.

The cofinality of any ordinal <math>\alpha</math> is a regular ordinal, that is, the cofinality of the cofinality of <math>\alpha</math> is the same as the cofinality of <math>\alpha.</math>  So the cofinality operation is [[idempotent]].

==Cofinality of cardinals==

If <math>\kappa</math> is an infinite cardinal number, then <math>\operatorname{cf}(\kappa)</math> is the least cardinal such that there is an [[bounded (set theory)|unbounded]] function from <math>\operatorname{cf}(\kappa)</math> to <math>\kappa;</math> <math>\operatorname{cf}(\kappa)</math> is also the cardinality of the smallest set of strictly smaller cardinals whose sum is <math>\kappa;</math> more precisely
<math display=block>\operatorname{cf}(\kappa) = \min \left\{ |I|\ :\ \kappa = \sum_{i \in I} \lambda_i\ \land \forall i \in I \colon \lambda_i < \kappa\right\}.</math>

That the set above is nonempty comes from the fact that
<math display=block>\kappa = \bigcup_{i \in \kappa} \{i\}</math>
that is, the [[disjoint union]] of <math>\kappa</math> singleton sets. This implies immediately that <math>\operatorname{cf}(\kappa) \leq \kappa.</math> 
The cofinality of any totally ordered set is regular, so <math>\operatorname{cf}(\kappa) = \operatorname{cf}(\operatorname{cf}(\kappa)).</math>

Using [[König's theorem (set theory)|König's theorem]], one can prove <math>\kappa < \kappa^{\operatorname{cf}(\kappa)}</math> and <math>\kappa < \operatorname{cf}\left(2^\kappa\right)</math> for any infinite cardinal <math>\kappa.</math>

The last inequality implies that the cofinality of the cardinality of the continuum must be uncountable. On the other hand,
<math display=block>\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 display=block>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 <math>\delta</math>
<math display=block>\operatorname{cf} (\aleph_\delta) = \operatorname{cf} (\delta).</math>

On the other hand, if the [[axiom of choice]] holds, then for a successor or zero ordinal <math>\delta</math>
<math display=block>\operatorname{cf} (\aleph_\delta) = \aleph_\delta.</math>

==See also==

* {{annotated link|Club set}}
* {{annotated link|Initial ordinal}}

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

{{Order theory}}

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