By using the Löwenheim–Skolem theorem & Mostowski collapse, in every model $V$ of $ZF+Con(ZF)$ there is a countable transitive set $M$ such that $(M,\in\_M) \models ZF$. Is the following "converse" true?



> 
> In every model $V$ of $ZF$ and every transitive set $M \in V$ such that $(M,\in\_M) \models ZF$, there exists a transitive set $N \in V$ such that:
> 
> 
> 1. $M \in N$
> 2. $(N,\in\_N) \models ZF$
> 3. $M$ is countable inside $N$
> 
> 
> 



Reverse Skolem's paradox
The answer to your question is no. For example, suppose $\kappa$ is inaccessible and $M = V\_{\kappa}$. Then $M$ is a model of $\sf ZF$ but $M$ cannot be countable in $N$ as it is not countable in $V$.


Of course there will always be a set generic extension of $V$ containing such an $N$ (just generically add a bijection from $M$ to omega)


