Suppose there exists a transitive model of $\sf ZFC$. Is it the case that every consistent theory that extends $\sf ZF$ must have a model that is an element of the minimal transitive model of $\sf ZFC$?



Does every consistent extension of ZF have a model in the minimal transitive model of ZFC?
The answer is no, because by the Gödel-Rosser theorem, there are continuum many consistent completions of ZF, but the minimal transitive model of ZFC is countable, and so has only countably many theories. So some of the consistent extensions are not realized in that model.


Another argument is simply this: the theory of the minimal model itself is a consistent extension of ZF, but this theory cannot be an element of the minimal transitive model (that is, it cannot have its own theory as an element), because from this theory you can reconstruct the model itself---some of the sentences of the theory assert that there is an object satisfying a certain definition, and all that one would want to know about those objects is asserted as part of the theory. So if the theory of the minimal model were inside that model, then that model would be able to construct a copy of itself as an element, which is impossible by minimality.


