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Suppose $M$ is a model of $\sf ZF+\neg AC$ that is externally bijective to an element $k \in M$. Obviously if $j$ denotes such an external bijection, then it cannot be used in Separation and Replacement within $M$, because the range of $j$ from $\mathcal P^M(k)$ would be a subset of $k$ that is bijective to $\mathcal P^M(k)$ by $j^{-1}$, and using Separation one can easily recover the diagonal which would be an element of $M$, and thus $M$ would no longer be bijective to $k$.

Would it be safe to use $j$ in stratified instances of Separation and Replacement within $M$? Where "stratified" is defined after Quine with the addition that $j(x)$ and $x$ receive the same type during stratification.

The main context is whether the stratification criterion can provide immunity from paradoxes when internalizing external functions inside some models of $\sf ZF$. If that holds, then it may constitute a powerful tool, since we may use it in internalizing automorphisms, embeddings, etc..

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There is no such model. By stratified Replacement with j, there is an S such that
y∈S<-->∃x(x∈k∧y=j(x)).

By Separation there is a C such that y∈C<-->y∈S∧y∉y. Then C∈C<-->C∉C.

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    $\begingroup$ Hmmm..., so the culprit is j-replacement. if we instead use $j^-1$ in your definition then $S$ would be $M$ itself and this is clearly contradictory. I see, thanks. $\endgroup$
    – Zuhair Al-Johar
    Commented Sep 16 at 15:35

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