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[EDIT:] new edits put in bold.

Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I mean we can have models of a suitable weakening of $\sf ZFCA$ in which there exist sets that are not comparable to any pure set, like for example forbidding impure to pure set replacement, or even work in $\sf ZCA$, and so $\sf GCH$ is no longer a problem of the pure set world in those settings. So, we can formalize $\sf GCH$ as: given any infinite set $A$, there doesn't exist any set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$. Where strict super- and sub-numerous are defined in the usual manner after existence injections and non-existence of a bijection. My naïve expectation is that permutation models are to settle such issue? Can they? Or otherwise, is there a version of forcing that works with urelements?

The point is that independence of the $\sf GCH$ might admit a easier solution in that setting through using permutation models by showing it to fail for sets incomparable to pure sets.

A related yet more concentrated question is to check if permutation models (or other method) can solve what could be called "Impure Continuum Hypothesis", which is the same question posed above but in addition require $A$ to be incomparable to any pure set.

Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I mean if we can have models of $\sf ZFCA$ in which there exist sets that are not comparable to any pure set, and so $\sf GCH$ is no longer a problem of the pure set world in those settings. So, we can formalize $\sf GCH$ as: given any infinite set $A$, there doesn't exist any set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$. Where strict super- and sub-numerous are defined in the usual manner after existence injections and non-existence of a bijection. My naïve expectation is that permutation models are to settle such issue? Can they? Or otherwise, is there a version of forcing that works with urelements?

The point is that independence of the $\sf GCH$ might admit a easier solution in that setting through using permutation models by showing it to fail for sets incomparable to pure sets.

A related yet more concentrated question is to check if permutation models (or other method) can solve what could be called "Impure Continuum Hypothesis", which is the same question posed above but in addition require $A$ to be incomparable to any pure set.

[EDIT:] new edits put in bold.

Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I mean we can have models of a suitable weakening of $\sf ZFCA$ in which there exist sets that are not comparable to any pure set, like for example forbidding impure to pure set replacement, and so $\sf GCH$ is no longer a problem of the pure set world in that setting. So, we can formalize $\sf GCH$ as: given any infinite set $A$, there doesn't exist any set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$. Where strict super- and sub-numerous are defined in the usual manner after existence injections and non-existence of a bijection. My naïve expectation is that permutation models are to settle such issue? Can they? Or otherwise, is there a version of forcing that works with urelements?

The point is that independence of the $\sf GCH$ might admit a easier solution in that setting through using permutation models by showing it to fail for sets incomparable to pure sets.

A related yet more concentrated question is to check if permutation models (or other method) can solve what could be called "Impure Continuum Hypothesis", which is the same question posed above but in addition require $A$ to be incomparable to any pure set.

[EDIT:] new edits put in bold.

Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I mean we can have models of a suitable weakening of $\sf ZFCA$ in which there exist sets that are not comparable to any pure set, like for example forbidding impure to pure set replacement, or even work in $\sf ZCA$, and so $\sf GCH$ is no longer a problem of the pure set world in those settings. So, we can formalize $\sf GCH$ as: given any infinite set $A$, there doesn't exist any set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$. Where strict super- and sub-numerous are defined in the usual manner after existence injections and non-existence of a bijection. My naïve expectation is that permutation models are to settle such issue? Can they? Or otherwise, is there a version of forcing that works with urelements?

The point is that independence of the $\sf GCH$ might admit a easier solution in that setting through using permutation models by showing it to fail for sets incomparable to pure sets.

A related yet more concentrated question is to check if permutation models (or other method) can solve what could be called "Impure Continuum Hypothesis", which is the same question posed above but in addition require $A$ to be incomparable to any pure set.

Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I mean we can have models of $\sf ZFCA$ in which there exist sets that are not comparable to any pure set, and so $\sf GCH$ is no longer a problem of the pure set world in that setting. So, we can formalize $\sf GCH$ as: given any infinite set $A$, there doesn't exist any set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$. Where strict super- and sub-numerous are defined in the usual manner after existence injections and non-existence of a bijection. My naïve expectation is that permutation models are to settle such issue? Can they? Or otherwise, is there a version of forcing that works with urelements?

The point is that independence of the $\sf GCH$ might admit a easier solution in that setting through using permutation models by showing it to fail for sets incomparable to pure sets.

A related yet more concentrated question is to check if permutation models (or other method) can solve what could be called "Impure Continuum Hypothesis", which is the same question posed above but in addition require $A$ to be incomparable to any pure set.

[EDIT:] new edits put in bold.

Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I mean we can have models of a suitable weakening of $\sf ZFCA$ in which there exist sets that are not comparable to any pure set, like for example forbidding impure to pure set replacement, and so $\sf GCH$ is no longer a problem of the pure set world in that setting. So, we can formalize $\sf GCH$ as: given any infinite set $A$, there doesn't exist any set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$. Where strict super- and sub-numerous are defined in the usual manner after existence injections and non-existence of a bijection. My naïve expectation is that permutation models are to settle such issue? Can they? Or otherwise, is there a version of forcing that works with urelements?

The point is that independence of the $\sf GCH$ might admit a easier solution in that setting through using permutation models by showing it to fail for sets incomparable to pure sets.

A related yet more concentrated question is to check if permutation models (or other method) can solve what could be called "Impure Continuum Hypothesis", which is the same question posed above but in addition require $A$ to be incomparable to any pure set.