Editing Cofinality (section)

==Regular and singular ordinals==
{{Main|Regular cardinal}}

A '''regular ordinal''' is an ordinal that is equal to its cofinality. A '''singular ordinal''' is any ordinal that is not regular.

Every regular ordinal is the [[initial ordinal]] of a cardinal.  Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular. Assuming the axiom of choice, <math>\omega_{\alpha+1}</math> is regular for each <math>\alpha.</math>  In this case, the ordinals <math>0, 1, \omega, \omega_1,</math> and <math>\omega_2</math> are regular, whereas <math>2, 3, \omega_\omega,</math> and <math>\omega_{\omega \cdot 2}</math> are initial ordinals that are not regular.

The cofinality of any ordinal <math>\alpha</math> is a regular ordinal, that is, the cofinality of the cofinality of <math>\alpha</math> is the same as the cofinality of <math>\alpha.</math>  So the cofinality operation is [[idempotent]].