Editing Cofinality (section)

==Cofinality of ordinals and other well-ordered sets==

The '''cofinality of an ordinal''' <math>\alpha</math> is the smallest ordinal <math>\delta</math> that is the [[order type]] of a [[cofinal subset]] of <math>\alpha.</math>  The cofinality of a set of ordinals or any other [[well-ordered set]] is the cofinality of the order type of that set.

Thus for a [[limit ordinal]] <math>\alpha,</math> there exists a <math>\delta</math>-indexed strictly increasing sequence with limit <math>\alpha.</math>  For example, the cofinality of <math>\omega^2</math> is <math>\omega,</math> because the sequence <math>\omega \cdot m</math> (where <math>m</math> ranges over the natural numbers) tends to <math>\omega^2;</math> but, more generally, any countable limit ordinal has cofinality <math>\omega.</math>  An uncountable limit ordinal may have either cofinality <math>\omega</math> as does <math>\omega_\omega</math> or an uncountable cofinality.

The cofinality of 0 is 0. The cofinality of any [[successor ordinal]] is 1.  The cofinality of any nonzero limit ordinal is an infinite regular cardinal.