Regular cardinal

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An infinite cardinal number κ is called regular if cf(κ) = κ, where cf is the cofinality operation. This says that κ cannot be expressed as the union (supremum) of a collection of less than κ smaller cardinals. If we demand a regular cardinal be also an uncountable limit cardinal, we get a weakly inaccessible cardinal (<math>\aleph_0</math> is a regular limit cardinal, but it is explicitly not an inaccessible cardinal because it is countable).

Infinite cardinals which are not regular are called singular (the existence of singular cardinals requires the Axiom of replacement).