Regular and Singular Ordinals
A regular ordinal is an ordinal which is equal to its cofinality. A singular ordinal is any ordinal which 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, is regular for each α. In this case, the ordinals 0, 1, and are regular, whereas 2, 3, and ωω·2 are initial ordinals which are not regular.
The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation is idempotent.
Read more about this topic: Cofinality
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