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 a limit cardinal, we get an inaccessible cardinal.

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