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Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models T$ iff $X \models_{V_\kappa} T$, where $\models_{V_\kappa}$ is the relation on $V_\kappa$ obtained by relativizing the definition of satisfaction to $V_\kappa$ (not by restricting $\models$ to $V_\kappa$).

Question. Let $M$ denote a transitive model of ZFC. Suppose that for every second order theory $T \in M$ and every structure $X \in M$, we have that $X \models T$ iff $X \models_M T$, where $\models_{M}$ is the relation on $M$ obtained by relativizing the definition of satisfaction to $M$. Is $M$ necessarily equal to $V_\kappa$ for some inaccessible $\kappa$?

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No, not at all. But it does tell you something about $M$. It tells you that it knows about sufficiently many of the subsets of $X$.

So if, for example, $M=V_\kappa$ for a worldly cardinal which is not inaccessible, then this is also true. For obvious reasons: if $X\in M$ then $\mathcal P(X)\in M$, so every possible subset is in $M$ and $M$ and $V$ agree about second-order (and even a bit higher than that) statement regarding $X$.

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    $\begingroup$ Thanks. Any ideas whether or not the property of interest implies that $M=V_\kappa$ for some cardinal $\kappa$? $\endgroup$ Jul 26, 2014 at 21:33
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    $\begingroup$ It seems plausible. I have been thinking about this for a bit, but I'm not 100% sure yet. $\endgroup$
    – Asaf Karagila
    Jul 26, 2014 at 21:37

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