Cofinality - Cofinality of Ordinals and Other Well-ordered Sets

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 ωω 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.

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