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This question says that the theory ZFC + $0^\#$ has incomparable minimal transitive models. It proves this as follows my emphasis):

[F]or every c.e. $T⊢\text{ZFC\P}+0^\#$ having a model $M$ with $On^M = α < ω_1$, the intersection of all such $M$ equals $L_α$, and furthermore a subset of $L_α$ is definable (with parameters) in all such $M$ iff it is in $L_{α^{+,\mathrm{CK}}}$. To see this (briefly), $0^\#$ allows $M$ to 'continue' $L$ beyond $α$, and $L_{α^{+,\mathrm{CK}}}⊆(L_{α^{+,\mathrm{CK}}})^M$ (the well-founded part of any model of KP being admissible), and so $L_{α^{+,\mathrm{CK}}}∩V_α=L_α$. Also, existence of $M$ is $Σ^1_1(α)$, so the intersection of all $M$ is at most $L_{α^{+,\mathrm{CK}}}$.

The bolded part uses Mostowski's absoluteness theorem. But to use Mostowski's absoluteness theorem we need $L_{α^{++,\mathrm{CK}}}$ to see that $\alpha$ is countable, and I have no idea how to prove that.

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    $\begingroup$ I recommend you cross-post your question on MO if you do not receive an answer within a week. $\endgroup$
    – Hanul Jeon
    Commented Jun 21 at 4:00
  • $\begingroup$ I second @HanulJeon's suggestion; I think this would be a fine fit for MO (and if you'd rather not ask it there I'd be happy to). $\endgroup$ Commented Jul 5 at 4:09
  • $\begingroup$ Are you asking the question in the title or how to prove the statement in bold in the body of the question? And in the title, should the theory be r.e.? $\endgroup$
    – Farmer S
    Commented Jul 6 at 22:45

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I copied this question to MathOverflow, and a comment by Farmer S to the MathOverflow version linked to this answer by Farmer S to MathOverflow question, which proves that if $\alpha$ is the least ordinal that is the height of a transitive model of $\text{ZFC}+0^\#$ and $\beta$ is the least admissible ordinal greater than $\alpha$, then there is such a model coded by a set in $L_{\beta+1}$. The proof seems to work for every theory extending $\text{ZFC}+0^\#$.

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