In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality is a regular cardinal if and only if every unbounded. Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.

In the presence of the axiom of choice, any cardinal number can be well-ordered, and so the following are equivalent:

1. is a regular cardinal.

2. .

3. .

4. The category and all functions between them is closed under colimits.

5. is a regular ordinal (see below).

Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts.

The situation is slightly more complicated in contexts where the axiom of choice might fail, as in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above equivalence holds for well-orderable cardinals only.

An infinite ordinal is a regular ordinal if it is a limit ordinal that is not the limit of a set of smaller ordinals that as a set has order type. A regular ordinal is always an initial ordinal (see the example below).

Examples

(aleph-null, 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; therefore it is singular.

is the next cardinal number are countable cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.

is the first infinite cardinal that is singular (the first infinite ordinal, and the first infinite limit ordinal). Proving the existence of singular cardinals requires the axiom of replacement in Zermelo set theory is what led Fraenkel to postulate this axiom.

Uncountable (weak) limit cardinals that are also regular are known as (weakly) inaccessible cardinals. They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom. Inaccessible cardinals are necessarily fixed pointss of the aleph function and is therefore singular.

Properties

If the axiom of choice holds, then every successor cardinal is regular. Thus the regularity or singularity of most aleph numbers can be checked depending on whether the cardinal is a successor cardinal or a limit cardinal. Some cardinalities cannot be proven to be equal to any particular aleph, for instance the cardinality of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see Easton's theorem). The continuum hypothesis, which is regular assuming choice.

Without the axiom of choice: there would be cardinal numbers that were not well-orderable. Moreover, the cardinal sum of an arbitrary collection could not be defined. Therefore, only the aleph numbers could meaningfully be called regular or singular cardinals.Furthermore, a successor aleph would need not be regular. For instance, the union of a countable set of countable sets would not necessarily be countable. It is consistent with ZF be the limit of a countable sequence of countable ordinals as well as the set of real numbers be a countable union of countable sets. is singular (a result proved by Moti Gitik).

-elementary embeddings is club.

a small embedding.Corollary 2.2

See also

• Inaccessible cardinal

References

• , Elements of Set Theory,

• , Set Theory, An Introduction to Independence Proofs,