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This is probably a pretty simple question, but I'm tying myself in knots over it.
We're all familiar with the Reflection Theorem, Lowenheim-Skolem Theorem, and Mostowski Collapse Lemma for getting countable transitive models of finite fragments of ZFC. You take a finite fragment $\Gamma$, use the reflection theorem to get that $\Gamma$ holds in some $V_\kappa$, Skolemise over $V_\kappa$, and then Collapse to a ctm.
My question. Is there (or could there be) a countable transitive model satisfying the same first-order truths as $V$? Obviously, by Tarski, such a countable transitive model is not first-order definable.
A possible route:
Suppose full second-order reflection is true of $V$. Let $A$ be a second-order parameter that contains a witness for every parameter-free sentence of first-order $ZFC$ true in $V$. Then, just reflect the sentence $x \in A$, Skolemise, and Collapse as normal (I feel like I might be pulling a fast-one here).