Keisler extension property

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 $\kappa$.