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Can independence of $\sf CH$ from $\sf ZFCA$ be established using $\sf FM $ permutation models? And if so, then historically did this came first or Cohen's forcing?

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No, it cannot. This is because CH is fundamentally a statement about pure sets (= sets not containing urelements in their transitive closures). No independence result for statements about "pure sets" can be established by permutation models, since if ${\bf V}$ is a permutation submodel of ${\bf W}$ then ${\bf V}$ and ${\bf W}$ have the same pure sets.

That said, this same issue does not apply to GCH, at least if phrased loosely as "If $A,B$ are infinite sets and there are injections from $A$ to $B$ and from $B$ to $\mathcal{P}(A)$, then $B$ is equinumerous with either $A$ or $\mathcal{P}(A)$."

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  • $\begingroup$ But why one should stick to this pure set exposition of the problem? Why not if $A$ is a countable set, then there is no set $B$ that is strictly supernumerous to $A$ and strictly subnumerous to $\mathcal P(A)$. This statement is not restricted to pure sets, and it seems to be the heart of the issue? Can a permutation model solve this? $\endgroup$
    – Zuhair Al-Johar
    Commented May 11 at 18:25
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    $\begingroup$ The issue is not whether you can formulate CH in terms of atoms; the issue is that CH (however you rephrase it) is equivalent to a statement about pure sets. So its truth value cannot be changed without changing the pure part of the universe. $\endgroup$
    – Gabe Goldberg
    Commented May 11 at 18:36
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    $\begingroup$ @GabeGoldberg, Ah I see. This is better phrased. So, what you and Noah are saying is that if a statement is equivalent to a statement about pure sets, then it cannot be solved by a permutation model, right! $\endgroup$
    – Zuhair Al-Johar
    Commented May 11 at 18:39
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    $\begingroup$ @NoahSchweber Don't Dedekind-finite infinite sets already violate the "loosely phrased" GCH? $\endgroup$
    – bof
    Commented May 11 at 19:11
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    $\begingroup$ @bof Yup! They do indeed (and a lot of other things too). $\endgroup$
    – Noah Schweber
    Commented May 11 at 19:12

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