In a [posting](https://math.stackexchange.com/questions/4206809/is-countability-absolute-over-supertransitive-models-of-set-theory?noredirect=1#comment8734837_4206809) to mathstackexchange I've alluded to the concept of supertransitive model. Now $M$ is a supertransitive model of a set $Q$ of first order sentences, denoted by $M \models^{sptr} Q$, is defined as $M$ being a set obeying $\sf ZFC$ rules and that is supertransitive (i.e.; transitive and has all subsets of all of its elements being among it elements too) and that satisfy $Q$.


Now every $V\_\kappa$ where $\kappa$ is a wordly cardinal, would serve as a supertransitive model of $\sf ZFC$, and apparantly there is no upper limit to that. However, is there a lower bound on the size of a supertransitive model of a theory? Especially of $\sf ZFC$ itself, i.e. is there a lower bound on the cardinality of a supertranstive model $M$ of $\sf ZFC$?



Is there a lower bound on the size of a supertransitive model of ZFC?
If $M$ is supertransitive and satisfies $\sf ZFC$, then $\omega\in M$, and more importantly, $V\_\omega\in M$.


Now by recursion, if $\alpha$ is an ordinal in $M$, then $V\_\alpha\in M$ as well.


Therefore $M$ must agree on the $V\_\alpha$ hierarchy, and therefore it must have the form $V\_\kappa$ for a worldly cardinal $\kappa$, or $M=V$.


So the smallest supertransitive model is the least worldly cardinal.


