Regular cardinal

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that ${\displaystyle \kappa }$ is a regular cardinal if and only if every unbounded subset ${\displaystyle C\subseteq \kappa }$ has cardinality ${\displaystyle \kappa }$. Infinite well-ordered 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 ${\displaystyle \kappa }$:

1. ${\displaystyle \kappa }$ is a regular cardinal.
2. If ${\displaystyle \kappa =\sum _{i\in I}\lambda _{i}}$ and ${\displaystyle \lambda _{i}<\kappa }$ for all ${\displaystyle i}$, then ${\displaystyle |I|\geq \kappa }$.
3. If ${\displaystyle S=\bigcup _{i\in I}S_{i}}$, and if ${\displaystyle |I|<\kappa }$ and ${\displaystyle |S_{i}|<\kappa }$ for all ${\displaystyle i}$, then ${\displaystyle |S|<\kappa }$.
4. The category ${\displaystyle \operatorname {Set} _{<\kappa }}$ of sets of cardinality less than ${\displaystyle \kappa }$ and all functions between them is closed under colimits of cardinality less than ${\displaystyle \kappa }$.
5. ${\displaystyle \kappa }$ 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 ${\displaystyle \alpha }$ 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 ${\displaystyle \alpha }$. A regular ordinal is always an initial ordinal, though some initial ordinals are not regular, e.g., ${\displaystyle \omega _{\omega }}$ (see the example below).