The [question](https://mathoverflow.net/questions/437643/does-every-consistent-extension-of-zf-have-a-model-in-the-minimal-transitive-mod) of whether the minimal transitive model $M$ of $\sf ZFC$ have a model of each consistent extension $\sf T$ of $\sf ZF$, is [answered](https://mathoverflow.net/a/437644/95347) to the negative, basically because $M$ is countable and we have $2^\omega$ many theories $\sf T$ extending $\sf ZF$.


Now I have three responses to that:



> 
> Can $M$ have a model of each consistent *parameter free definable* theory $\sf T$ that extends $\sf ZF$, among its elements? I know that this is true for all such theories if they are elements of $M$ (see [answer](https://mathoverflow.net/a/437650/95347))! Here the question includes even those definable outside of $M$?
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> Can every first order theory that consistently extend $\sf ZF$ by *finitely* many axioms, have a model in $M$?
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> Can every theory that consistently extends $\sf ZF$, have a model in $\mathcal P(M)$?
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Is there a model of each of the following kinds of theories in the first transitive model of ZFC?
The answer to question 1 is no, because the theory of the minimal transitive model itself is parameter-free definable as that theory, and in the other answer I explained why this theory is not an element of the minimal model. More generally, one cannot in general make much of a conclusion about an object from it being definable, becauase in the light of the universal definition, any object can in principle be made definable in a forcing extension.


The answer to the second question is yes. Because the minimal transitive model of ZFC is transitive, it has the true $\omega$ and thus it has the true ZFC and all finite extensions, and it agrees on consistency statements since it has all the same proofs that we do.


I don't understand the third question (or your remarks about subsets on the other post). I'm not sure about what kind of models you intend---do you intend submodels of the minimal model $\langle L\_\alpha,\in\rangle$, that is, with the same $\in$ relation? All such models would be well-founded and thus would have transitive collapses either to a set in $L\_\alpha$ or to $L\_\alpha$ itself.


