*Throughout assume $\mathsf{ZFC}$ + "There is a proper class of inaccessible cardinals." I'm also happy to strengthen the large cardinal hypothesis if that would help.*

Say that a model $M\models\mathsf{ZFC}$ is **powerful** iff every end extension satisfying $\mathsf{ZFC}$ is a top extension. The transitive powerful models are exactly the $V_\alpha$s where $\alpha$ is a worldly cardinal: that every $V_{\mathsf{worldly}}$ is powerful is trivial, and the large cardinal hypothesis above gives the other half of the result.

The ill-founded situation is on the other hand completely unclear to me:

Is there an ill-founded powerful model of $\mathsf{ZFC}$?

I don't really see where to start with this. Certainly there are no *countable* powerful models of $\mathsf{ZFC}$, ill-founded or otherwise, since we can (genuinely) force over such models; however, I don't see any useful tools for analyzing uncountable ill-founded models. The existence of rigid ill-founded models *(e.g. the substructure of any ill-founded model of $\mathsf{ZFC+V=L}$ consisting of the parameter-freely-definable elements)* does make the existence of powerful models feel not *totally* implausible, but that said I don't see how any of the techniques for building examples of the former type are useful for attempting to build an example of the latter.

noend extensions? $\endgroup$4more comments