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 subset
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 then the following are equivalent for a cardinal
: 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.
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 less than
(aleph-null) is a regular cardinal because its initial ordinal,
is the next ordinal number greater than
It is singular, since it is not a limit ordinal.
is the next cardinal number greater than
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
Assuming the axiom of choice,
, and the first infinite limit ordinal that 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.
[1] Uncountable (weak) limit cardinals that are also regular are known as (weakly) inaccessible cardinals.
Inaccessible cardinals are necessarily fixed points of the aleph function, though not all fixed points are regular.
For instance, the first fixed point is the limit of the
If the axiom of choice holds, then every successor cardinal is regular.
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).
Without the axiom of choice: there would be cardinal numbers that were not well-orderable.
[citation needed] Moreover, the cardinal sum of an arbitrary collection could not be defined.
[citation needed] Therefore, only the aleph numbers could meaningfully be called regular or singular cardinals.
[citation needed] Furthermore, it is consistent with ZF when not including AC that every aleph bigger than
is singular (a result proved by Moti Gitik).
is regular iff the set of
is uncountable and regular iff there is an