Weakly inaccessible cardinal

In mathematics, a cardinal number κ > ‭א‬₀ is called weakly inaccessible iff the following two conditions hold.

1. cf(κ) = κ, 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 λ < κ, there is another cardinal number between λ and κ. 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 weakly inaccessible cardinals provably cannot be proven in ZFC.

See also

• Strongly inaccessible cardinal
• Large cardinal