Weakly inaccessible cardinal
, a cardinal number
κ > א0
is called weakly inaccessible iff
the following two conditions hold.
- cf(κ) = κ, where cf denotes the cofinality. Such a cardinal is called a regular cardinal.
- 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
, the existence of weakly inaccessible cardinals provably
cannot be proven in ZFC.