Editing Regular cardinal (section)

== 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>