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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.

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  • $\begingroup$ A set theory with Choice that permits incomparable cardinalities just seems to be missing an axiom. $\endgroup$ Commented May 26 at 12:33
  • $\begingroup$ Yes! You are right. The question should be about ZCA. Anyhow. $\endgroup$ Commented May 26 at 17:35
  • $\begingroup$ Even Z (without foundation) is enough to prove the standard equivalences of choice, e.g. AC iff Well-ordering Thm iff cardinal trichotomy iff surjective cardinal trichotomy. $\endgroup$ Commented May 28 at 8:25
  • $\begingroup$ @ElliotGlazer Yes! I should have added that the incomparability to pure sets condition must be changed to non-subnumerosity to pure sets. $\endgroup$ Commented May 28 at 9:13

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In ZFCA, every set is equinumerous with a pure set, since every well ordering of a set makes it bijective with an ordinal, which is pure. Therefore the cardinal structure of any model of ZFCA is reflected in its pure sets, and in particular, CH and the GCH will hold, respectively, if and only if they hold in the underlying ZFC model of pure sets.

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  • $\begingroup$ In reality the original question I had in mind was in relation to ZCA where I thought this incomparability can occur. Anyhow I think the underlying ZFA theory for FM permutation models have a replacement schema so the question might not be relevant to them. But anyhow. How can we prove matters in the setting of ZCA or even in ZFCA but with forbidding impure to pure replacement. I mean those I think can permit such incomparabilities, so how to settle such questions in these contexts. That's the essence of the question really. $\endgroup$ Commented May 26 at 17:40
  • $\begingroup$ I've edited the question to meet the original intention that motivated it. $\endgroup$ Commented May 26 at 17:49
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    $\begingroup$ Ah, OK, in that case I've lost interest... $\endgroup$ Commented May 26 at 19:46
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    $\begingroup$ @ZuhairAl-Johar If you have a materially different question from the one you originally posed, then the etiquette is that you create a new MO question, rather than edit your question, which can make it seem that the correct answers to your original question are wrong or misguided. $\endgroup$ Commented May 27 at 13:39
  • $\begingroup$ @TimothyChow, OK good suggestion. Thanks $\endgroup$ Commented May 27 at 14:00

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