It is known that Morse-Kelly (MK) set theory forms a metatheory for ZFC. For example:


MK proves Con(ZFC). In fact, Joel David Hamkins claims in his blog post "Kelly-Morse set theory implies Con(ZFC) and much more" that in MK "there is a transitive model of ZFC, and furthermore that the universe $V$ is the union of an elementary chain of elementary rank initial segments $V\_\theta$ of $V$, each of which , in particular, is a transitive model of ZFC."


My question is this: If in MK there is a transitive model $M$ of ZFC, is there in MK the generic extension $M$[$G$], where the generic $G$ is definable in MK? Also, if so, is there a $M$[$G$] in MK in which cardinals are collapsed?



A Question Regarding Defining Generic Extensions of ZF and ZFC in Morse-Kelly Set Theory
If there is a transitive model of ZFC, then there is a smallest ordinal $\alpha$ such that $L\_\alpha$ is a model of ZFC, and this is called the minimal transitive model of ZFC, because it is contained in all others. So we can define this model in the theory KM. Since it is countable in $L$, furthermore, there will be $L\_\alpha$-generic filters $G\in L$ for any particular poset $\mathbb{P}\in L\_\alpha$, and so there will be an $L$-least such $G$ in $L$ that is $L\_\alpha$-generic. Thus, we can also define all kinds of forcing extensions $L\_\alpha[G]$, for any particular $\mathbb{P}\in L\_\alpha$, since all elements of $L\_\alpha$ are definable there without parameters and hence also definable in KM. So we arrive in this way at numerous models $L\_\alpha[G]$ that are definable in KM. By taking $\mathbb{P}$ to be the forcing to collapse $\omega\_1$ to $\omega$ (from the perspective of $L\_\alpha$), these extensions will also collapse cardinals of $L\_\alpha$. 


