Editing Regular cardinal

{{Short description|Type of cardinal number in mathematics}}
In [[set theory]], a '''regular cardinal''' is a  [[cardinal number]] that is equal to its own [[cofinality]].  More explicitly, this means that <math>\kappa</math> is a regular cardinal if and only if every [[cofinal (mathematics)|unbounded]] subset <math>C \subseteq \kappa</math> has cardinality <math>\kappa</math>.  Infinite [[well-order]]ed cardinals that are not regular are called '''singular cardinals'''. Finite cardinal numbers are typically not called regular or singular.

In the presence of the [[axiom of choice]], any cardinal number can be well-ordered, and then the following are equivalent for a cardinal <math>\kappa</math>:
# <math>\kappa</math> is a regular cardinal.
# If <math>\kappa = \sum_{i \in I} \lambda_i</math> and <math>\lambda_i < \kappa</math> for all <math>i</math>, then <math>|I| \ge \kappa</math>.
# If <math>S = \bigcup_{i \in I} S_i</math>, and if <math>|I| < \kappa</math> and <math>|S_i| < \kappa</math> for all <math>i</math>, then <math>|S| < \kappa</math>. That is, every union of fewer than <math>\kappa</math> sets smaller than <math>\kappa</math> is smaller than <math>\kappa</math>.
# The [[category (mathematics)|category]] <math>\operatorname{Set}_{<\kappa}</math> of sets of cardinality less than <math>\kappa</math> and all functions between them is closed under [[colimit]]s of cardinality less than <math>\kappa</math>.
# <math>\kappa</math> is a regular ordinal (see below).
Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts.

The situation is slightly more complicated in contexts where the [[axiom of choice]] might fail, as in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above equivalence holds for well-orderable cardinals only.

An infinite [[ordinal number|ordinal]] <math>\alpha</math> is a '''regular ordinal''' if it is a [[limit ordinal]] that is not the limit of a set of smaller ordinals that as a set has [[order type]] less than <math>\alpha</math>. A regular ordinal is always an [[initial ordinal]], though some initial ordinals are not regular, e.g., <math>\omega_\omega</math> (see the example below).

== Examples ==

The ordinals less than <math>\omega</math> are finite. A finite sequence of finite ordinals always has a finite maximum, so <math>\omega</math> cannot be the limit of any sequence of type less than <math>\omega</math> whose elements are ordinals less than <math>\omega</math>, and is therefore a regular ordinal. <math>\aleph_0</math> ([[aleph-null]]) is a regular cardinal because its initial ordinal, <math>\omega</math>, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite.

<math>\omega+1</math> is the [[successor ordinal|next ordinal number]] greater than <math>\omega</math>. It is singular, since it is not a limit ordinal. <math>\omega+\omega</math> is the next limit ordinal after <math>\omega</math>. It can be written as the limit of the sequence <math>\omega</math>, <math>\omega+1</math>, <math>\omega+2</math>, <math>\omega+3</math>, and so on. This sequence has order type <math>\omega</math>, so <math>\omega+\omega</math> is the limit of a sequence of type less than <math>\omega+\omega</math> whose elements are ordinals less than <math>\omega+\omega</math>; therefore it is singular.

<math>\aleph_1</math> is the [[successor cardinal|next cardinal number]] greater than <math>\aleph_0</math>, so the cardinals less than <math>\aleph_1</math> are [[countable set|countable]] (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So <math>\aleph_1</math> cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.

<math>\aleph_\omega</math> is the next cardinal number after the sequence <math>\aleph_0</math>, <math>\aleph_1</math>, <math>\aleph_2</math>, <math>\aleph_3</math>, and so on. Its initial ordinal <math>\omega_\omega</math> is the limit of the sequence <math>\omega</math>, <math>\omega_1</math>, <math>\omega_2</math>, <math>\omega_3</math>, and so on, which has order type <math>\omega</math>, so <math>\omega_\omega</math> is singular, and so is <math>\aleph_\omega</math>. Assuming the axiom of choice, <math>\aleph_\omega</math> is the first infinite cardinal that is singular (the first infinite ''ordinal'' that is singular is <math>\omega+1</math>, and the first infinite ''limit ordinal'' that is singular is <math>\omega+\omega</math>). Proving the existence of singular cardinals requires the [[axiom schema of replacement|axiom of replacement]], and in fact the inability to prove the existence of <math>\aleph_\omega</math> in [[Zermelo set theory]] is what led [[Adolf Abraham Halevi Fraenkel|Fraenkel]] to postulate this axiom.<ref>{{citation
 | last = Maddy | first = Penelope | authorlink = Penelope Maddy
 | doi = 10.2307/2274520
 | jstor = 2274520
 | issue = 2
 | journal = [[Journal of Symbolic Logic]]
 | mr = 947855
 | pages = 481–511
 | quote =  Early hints of the Axiom of Replacement can be found in Cantor's letter to Dedekind [1899] and in Mirimanoff [1917]
 | title = Believing the axioms. I
 | volume = 53
 | year = 1988}}. Maddy cites two papers by Mirimanoff, "Les antinomies de Russell et de Burali-Forti et le problème fundamental de la théorie des ensembles" and "Remarques sur la théorie des ensembles et les antinomies Cantorienne", both in ''L'Enseignement Mathématique'' (1917).</ref>

Uncountable (weak) [[limit cardinal]]s that are also regular are known as (weakly) [[inaccessible cardinals]].  They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom.  Inaccessible cardinals are necessarily [[fixed point (mathematics)|fixed point]]s of the [[aleph number|aleph function]], though not all fixed points are regular.  For instance, the first fixed point is the limit of the <math>\omega</math>-sequence <math>\aleph_0, \aleph_{\omega}, \aleph_{\omega_{\omega}}, ...</math> and is therefore singular.

== Properties ==

If the [[axiom of choice]] holds, then every [[successor cardinal]] is regular.  Thus the regularity or singularity of most aleph numbers can be checked depending on whether the cardinal is a successor cardinal or a limit cardinal.  Some cardinalities cannot be proven to be equal to any particular aleph, for instance the [[cardinality of the continuum]], whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see [[Easton's theorem]]). The [[continuum hypothesis]] postulates that the cardinality of the continuum is equal to <math>\aleph_1</math>, which is regular assuming choice.

Without the axiom of choice: there would be cardinal numbers that were not well-orderable. {{citation needed|date=August 2024}} Moreover, the cardinal sum of an arbitrary collection could not be defined.{{citation needed|date=August 2024}} Therefore, only the [[aleph number]]s could meaningfully be called regular or singular cardinals.{{citation needed|date=August 2024}}Furthermore, a successor aleph would need not be regular. For instance, the union of a countable set of countable sets would not necessarily be countable. It is consistent with [[Zermelo–Fraenkel set theory|ZF]] that <math>\omega_1</math> be the limit of a countable sequence of countable ordinals as well as the set of [[real number]]s be a countable union of countable sets.{{citation needed|date=August 2024}} Furthermore, it is consistent with ZF when not including AC that every aleph bigger than <math>\aleph_0</math> is singular (a result proved by [[Moti Gitik]]).

If <math>\kappa</math> is a limit ordinal, <math>\kappa</math> is regular iff the set of <math>\alpha<\kappa</math> that are critical points of <math>\Sigma_1</math>-[[Elementary embedding|elementary embeddings]] <math>j</math> with <math>j(\alpha)=\kappa</math> is [[Club set|club]] in <math>\kappa</math>.<ref>T. Arai, "Bounds on provability in set theories" (2012, p.2). Accessed 4 August 2022.</ref>

For cardinals <math>\kappa<\theta</math>, say that an elementary embedding <math>j:M\to H(\theta)</math> a ''small embedding'' if <math>M</math> is transitive and <math>j(\textrm{crit}(j))=\kappa</math>. A cardinal <math>\kappa</math> is uncountable and regular iff there is an <math>\alpha>\kappa</math> such that for every <math>\theta>\alpha</math>, there is a small embedding <math>j:M\to H(\theta)</math>.<ref>Holy, Lücke, Njegomir, "[https://www.sciencedirect.com/science/article/pii/S0168007218301167 Small embedding characterizations for large cardinals]". Annals of Pure and Applied Logic vol. 170, no. 2 (2019), pp.251--271.</ref><sup>Corollary 2.2</sup>

== See also ==
* [[Inaccessible cardinal]]

==References==
{{reflist}}

* {{aut|[[Herbert Enderton|Herbert B. Enderton]]}}, ''Elements of Set Theory'', {{isbn|0-12-238440-7}}
* {{aut|[[Kenneth Kunen]]}}, ''Set Theory, An Introduction to Independence Proofs'', {{isbn|0-444-85401-0}}

{{Mathematical logic}}
{{Set theory}}

[[Category:Cardinal numbers]]
[[Category:Ordinal numbers]]