Editing Regular cardinal (section)

{{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).