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
Famous quotes containing the words regular and/or singular:
“They were regular in being gay, they learned little things that are things in being gay, they learned many little things that are things in being gay, they were gay every day, they were regular, they were gay, they were gay the same length of time every day, they were gay, they were quite regularly gay.”
—Gertrude Stein (1874–1946)
“Panurge was of medium stature, neither too large, nor too small ... and subject by nature to a malady known at the time as “Money-deficiency,”Ma singular hardship; nevertheless, he had sixty-three ways of finding some for his needs, the most honorable and common of which was by a form of larceny practiced furtively.”
—François Rabelais (1494–1553)