Weakly inaccessible cardinal ($\mathrm{cf}(\kappa )=\kappa \text{ and }\kappa \text{ is an uncountable limit cardinal}$)

A weakly inaccessible cardinal is an uncountable regular cardinal that is a limit cardinal. It need not be a strong limit cardinal. Under the Generalized continuum hypothesis, it is a strongly inaccessible cardinal, since every smaller infinite $\lambda$ satisfies $2^{\lambda }={\lambda }^{+}<\kappa$.

A weakly inaccessible cardinal $\kappa$ is weakly Mahlo if $\{\alpha <\kappa :\alpha =\mathrm{cf}(\alpha )\}$ is a stationary set. Intersecting this set with the club set of uncountable limit cardinals shows that the weakly inaccessible cardinals below $\kappa$ are stationary, hence unbounded.

Consistency of ZFC implies consistency of ZFC with the Generalized continuum hypothesis and no weakly inaccessible cardinals. Pass to the constructible universe. If it has an inaccessible, cut at its least one; that rank segment still models ZFC and the Generalized continuum hypothesis, but has no inaccessibles. In this model the singular cardinal enumeration is continuous at every nonzero limit ordinal index. This is a relative-consistency construction, not a deduction of a transitive model from bare consistency.