Singular cardinal

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one that cannot be broken into a smaller collection of smaller parts.

If the axiom of choice holds (so that any cardinal number can be well-ordered), an infinite cardinal ${\displaystyle \kappa }$ is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than ${\displaystyle \kappa }$, the elements of which are cardinals less than ${\displaystyle \kappa }$. 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 definition is restricted to well-orderable cardinals only.

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