Inaccessible cardinal

In mathematics, a cardinal number k > <math>\aleph_0<math> (aleph-null) is called weakly inaccessible, or just inaccessible, if the following two conditions hold.

1. cf(k) = k, where cf denotes the cofinality. Such a cardinal is called a regular cardinal.
2. There is no next smaller cardinal number; i.e., for every cardinal l < k, there is another cardinal number between l and k. Such a cardinal number is called a limit cardinal.

Every transfinite cardinal number is either regular or a limit; however, only a rather large cardinal number can be both. In fact, assuming that ZFC is consistent, the existence of inaccessible cardinals provably cannot be proven in ZFC.

All inaccessible cardinals are large cardinals. Some inaccessible cardinals may be strongly inaccessible cardinals.