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 which 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 is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than, the elements of which are cardinals less than . (The situation is slightly more complicated in contexts where the axiom of choice might fail; 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.)
An infinite ordinal is regular if and only if it is a limit ordinal which is not the limit of a set of smaller ordinals which set has order type less than . A regular ordinal is always an initial ordinal, though some initial ordinals are not regular.
Infinite well-ordered cardinals which are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.
... Uncountable limit cardinals that are also regular are known as weakly inaccessible cardinals ... Inaccessible cardinals are necessarily fixed points of the aleph function, though not all fixed points are regular ... If the axiom of choice holds, then every successor cardinal is regular ...
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