*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.



Are there ill-founded "maximally wide" models of $\mathsf{ZFC}$?
Back in the mid 1980s I remember convincing myself (alas, in unpublished work) that *there is an ill-founded model $M$ of ZFC that has no end extension to another model of ZFC.* Such a model $M$ by default is powerful and technically answers the question.


My unpublished work above used techniques employed in the following related results from the same time period:


**1.** There are ill-founded models $M$ *of any given consistent extension of* ZFC that have no top extension to another model of ZFC; see Theorem 1.5 of [this paper of mine](https://www.ams.org/journals/tran/1984-283-02/S0002-9947-1984-0737894-1/S0002-9947-1984-0737894-1.pdf). Note that top extensions are referred to as rank extensions in the paper.


**2.** There are ill-founded models $M$ *of any given consistent extension of* ZFC such that there are no $N \supsetneq M$ such that $N$ is a *forcing* end extension of $M$. This was noted in [this other paper of mine](https://www.researchgate.net/publication/38382376_Conservative_Extensions_of_Models_of_Set_Theory_and_Generalizations#fullTextFileContent) (see the remark after Theorem 1.5, which itself relates to Theorem 1.4).


In (1) and (2) above, the cardinality of $M$ can be arranged to be $\aleph\_1$. This is optimum since every countable model of ZF (including ill-founded ones) has lots of top extensions and forcing extensions


