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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.

I originally posted this question on Stackexchange but two comments recommended moving it to MathOverflow.

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  • $\begingroup$ You didn't actually ask a question. (It seems there are two possible implied questions: (i) why does said theory have incomparable minimal models?, and (ii) why is the boldface statement true? Are you asking one and/or both of them (or something else)?) $\endgroup$
    – Farmer S
    Commented Jul 9 at 5:27
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    $\begingroup$ For (i): Let $\alpha$ be the least ordinal such that there is a transitive model $M$ of ZFC + "$0^\sharp$ exists" of ordinal height $\alpha$. (I am assuming such an ordinal $\alpha$ exists; so $\alpha$ is also countable.) Then there is also a model $N$ of height $\alpha$ satisfying the theory $T$, where $T$ is ZFC + "$0^\sharp$ exists" + $V=L[0^\sharp]$. Any two distinct models of $T$ of height $\alpha$ are ($\subseteq$)-incomparable (and in fact also incomparable in the sense of iterating)... $\endgroup$
    – Farmer S
    Commented Jul 9 at 6:00
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    $\begingroup$ ...For suppose $N$ is the unique such model. Then $N\in L_\beta$ where $\beta$ is the least admissible $>\alpha$. But $(0^\sharp)^N\notin L^N=L_\alpha$, so there is an ordinal $\gamma\in[\alpha,\beta)$ such that $L_\gamma$ projects to $\omega$. So there is a surjection $\pi:\omega\to\alpha$ with $\pi\in L_\beta$ (in fact $\pi\in L_{\gamma+1}$). But every set in $\mathcal{P}(\alpha) \cap L_\beta$ is definable from parameters over $N$, so $\pi$ is definable from parameters over $N$, contradicting that $N$ models ZFC. $\endgroup$
    – Farmer S
    Commented Jul 9 at 6:00
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    $\begingroup$ @HanulJeon It's Solovay's theorem that if $W\models$ ZFC and $W$ is transitive and $\mathbb{P}\in W$ and $A$ is a set of ordinals in $W[G]$ for every $(W,\mathbb{P})$-generic filter $G$, then $A\in W$. The same proof works here, replacing the collection of all $W$-generics with "sufficiently generic (over $L_\beta$)", which we can take to mean generic and such that $L_\beta[G]\models$ KP. (Here is a sketch of the proof: Take $(G_1,G_2)$ to be mutually generic with $G_1,G_2$ each sufficiently generic, and let $\sigma_1,\sigma_2\in L_\beta$ be names for $N$ w.r.t. $G_1,G_2$ respectively... $\endgroup$
    – Farmer S
    Commented Jul 11 at 8:17
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    $\begingroup$ @HanulJeon Because $N$ models $T$, which includes "$V=L[0^\sharp]$", it is naturally coded by a subset $N'$ of $\alpha$. And the height of $N$ is exactly $\alpha$, so since we have a code for $\alpha$ in $L_\beta[G]$, we can (with a $\Sigma_1$ formula) ask for both a code $N''$ for a model, and an isomorphism between $\alpha$ and the ordinals of $N''$. But $N'$ is the unique such code, so $N'\in L_\beta[G]$, so $N'\in L_\beta$. This gives $N\in L_\beta$, since, as you say, $N$ is externally wellfounded. (Actually the isomorphism between the ordinals exhibited by $N''$, with $\alpha$, is also.. $\endgroup$
    – Farmer S
    Commented Jul 12 at 10:44

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This answer by Farmer S to another question, which Farmer S linked in a comment to this question, 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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