Strongly inaccessible cardinal
In mathematics, a strongly inaccessible cardinal is an uncountable cardinal number κ that is regular and a strong limit cardinal.
In other words
- the cofinality cf(κ) = κ, and
- 2λ < κ for all λ < κ.
Assuming that ZFC is consistent, the existence of strongly inaccessible cardinals provably cannot be proved in ZFC. In fact, it cannot even be proved that the existence of strongly inaccessible cardinals is consistent with ZFC. Strongly inaccessible cardinals are therefore a type of large cardinal.
Under the Generalized Continuum Hypothesis, a cardinal is strongly inaccessible if and only if it is weakly inaccessible.
The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.