**Cofinality of Ordinals and Other Well-ordered Sets**

The **cofinality of an ordinal** α is the smallest ordinal δ which is the order type of a cofinal subset of α. 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, there exists a δ-indexed strictly increasing sequence with limit α. For example, the cofinality of ω² is ω, because the sequence ω·*m* (where *m* ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does ω_{ω} or an uncountable cofinality.

The cofinality of 0 is 0. The cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least ω.

Read more about this topic: Cofinality

### Famous quotes containing the word sets:

“The poem has a social effect of some kind whether or not the poet wills it to have. It has kinetic force, it *sets* in motion ... [ellipsis in source] elements in the reader that would otherwise be stagnant.”

—Denise Levertov (b. 1923)