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I know that if we assume zero sharp we obtain for a class of standard ordinals $L_\alpha\prec L_\beta$ , $\alpha <\beta$ and obviously $L_\alpha\in L_\beta$.

My question is : If we only assume the existence of a standard transitive model ,and from it we have a countable standard ordinal $\alpha$, $L_\alpha\models ZFC$, being for example $L_\alpha$ the minimal model,then there exist $N$ elementary end-extension $L_\alpha\prec N$ and $L_\alpha\in N$?, $N$ not necessarily externally well-founded.

Any help or reference would be appreciated.

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The answer is no, there can be no such elementary end-extension $N$ of the minimal model $L_\alpha$ with $L_\alpha\in N$. The reason is that $N$ would have to be $\omega$-standard, since it has the same $\omega$ as $L_\alpha$, and thus $N$ would have the correct understanding of the theory ZFC, and it would think that "there is a transitive model of ZFC". By elementarity, $L_\alpha$ would have to think that also, but it cannot since this would violate minimality.

This argument does not assume that $N$ is well-founded, but only that $L_\alpha$ is both an element and a transitive substructure of $N$.

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