Strongly inaccessible cardinal

A cardinal number κ > ‭א‬₀ is called strongly inaccessible iff the following conditions hold:

1. κ is weakly inaccessible; that is, cf(κ) = κ.
2. κ is a strong limit[?], that is, 2^λ < κ for all λ < κ.

Assuming that ZFC is consistent, the existence of strongly inaccessible cardinals provably cannot be proved in ZFC.