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    $\begingroup$ The completeness of the theories of real-closed fields and algebraically closed fields of characteristic $0$ are provable by elementary syntactic arguments (and in any case, they are arithmetical statements, hence automatically provable without AC). Thus, the problem is equivalent to showing that the field of algebraic numbers does not interpret a real-closed field. Since this field is countable, this should avoid issues with well orderability of $\mathbb R$. $\endgroup$ Commented Jul 13, 2021 at 13:35
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    $\begingroup$ Actually, why am I making it so complicated: the non-interpretability is itself an arithmetic statement, hence it is automatically provable without AC. $\endgroup$ Commented Jul 13, 2021 at 13:37
  • $\begingroup$ Emil, please post an answer! It may be good to explain in detail why the interpretation of those two structures amounts to the interpretability of their theories. I was worried about a model of ZF whose real numbers are very strange (e.g. countable union of countable sets etc.) and you are saying I don't need to worry about that at all. $\endgroup$ Commented Jul 13, 2021 at 13:41
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    $\begingroup$ I don't know about "much more", it's also reliant on absoluteness results. Mine is definitely cooler, though, since it involves forcing and dancing in two weddings with one tuchess. $\endgroup$ Commented Jul 13, 2021 at 13:58
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    $\begingroup$ Related, apparently. $\endgroup$ Commented Jul 16, 2021 at 18:18