Regular Cardinal - Examples

Examples

The ordinals less than are finite. A finite sequence of finite ordinals always has a finite maximum, so cannot be the limit of any sequence of type less than whose elements are ordinals less than, and is therefore a regular ordinal. (aleph-null) is a regular cardinal because its initial ordinal, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite.

is the next ordinal number greater than . It is singular, since it is not a limit ordinal. is the next limit ordinal after . It can be written as the limit of the sequence, and so on. This sequence has order type, so is the limit of a sequence of type less than whose elements are ordinals less than, therefore it is singular.

is the next cardinal number greater than, so the cardinals less than are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.

is the next cardinal number after the sequence, and so on. Its initial ordinal is the limit of the sequence, and so on, which has order type, so is singular, and so is . Assuming the axiom of choice, is the first infinite cardinal which is singular (the first infinite ordinal which is singular is ). Proving the existence of singular cardinals requires the axiom of replacement, and in fact the inability to prove the existence of in Zermelo set theory is what led Fraenkel to postulate this axiom.