Strongly inaccessible cardinal

A strongly inaccessible cardinal is an uncountable regular strong limit cardinal.

An inaccessible cardinal $\iota$ is an inaccessible limit of inaccessibles if the inaccessible cardinals below it are unbounded in $\iota$. A rank cutoff at the first inaccessible limit of inaccessibles shows that existence of unboundedly many inaccessibles does not prove existence of such a limit, provided the former theory is consistent.

If $\iota$ is the least inaccessible limit of inaccessibles, $V_{\iota }$ models ZFC and internally has unboundedly many inaccessible cardinals, while none of its ordinals is an inaccessible limit of inaccessibles. Rank agreement makes inaccessibility below $\iota$ absolute. If a model of unboundedly many inaccessibles has no such limit in the first place, no cutoff is needed. This proves the corresponding relative-consistency nonimplication.

An inaccessible cardinal $\kappa$ has the Keisler extension property when there is a proper transitive set $X⊋V_{\kappa }$ for whichSuch a $\kappa$ is not the least inaccessible cardinal: $X$ regards $\kappa$ as inaccessible, elementarity reflects the existence of an inaccessible into $V_{\kappa }$, and strong-inaccessibility absoluteness from rank agreement makes the resulting witness genuinely inaccessible below $\kappa$.