How questions similar to the continuum hypothesis can be solved if we work in a set theory that admits urelements and that permit impure sets that are incomparable to any pure set, for example $\sf ZCA$ or even a weakened version of $\sf ZFCA$ where replacement between the pure and impure sets is not allowed. So, to be specific how can we solve the question of independence of the Impure Continuum Hypothesis "ICH" from the rest of axioms of these theories? ICH can be stated as: for any impure infinite set $A$ that is not comparable to any pure set, there is no set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$? Is permutation models relevant to such contexts? Or there is a version of forcing that works with impure sets?

How questions similar to the continuum hypothesis can be solved if we work in a set theory that admits urelements and that permit impure sets that are not injective to any pure set, for example $\sf ZCA$ or even a weakened version of $\sf ZFCA$ where replacement between the pure and impure sets is not allowed. So, to be specific how can we solve the question of independence of the Impure Continuum Hypothesis "ICH" from the rest of axioms of these theories? ICH can be stated as: for any impure infinite set $A$ that is not injective to any pure set, there is no set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$? Is permutation models relevant to such contexts? Or there is a version of forcing that works with impure sets?