Regular Cardinal - Properties

Properties

Uncountable 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 points of the aleph function, though not all fixed points are regular. For instance, the first fixed point is the limit of the -sequence and is therefore singular.

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 cardinal numbers 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 postulates that the cardinality of the continuum is equal to which is regular.

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