Let $M$ be a countable transitive model of (enough of) ZFC. Mostowski's Absoluteness Theorem says that $\Pi^1_1$ statements are absolute between $M$ and larger models, in particular, between $M$ and the universe $V$.

For general $M$, this cannot be extended to $\Sigma^1_2$ statements, see Andrés Caicedo's answer here: Failure of Shoenfield's Absoluteness.

My question is: Can we have absoluteness of $\Sigma^1_2$ (and beyond) statements between $M$ and $V$ for some countable transitive models $M$? (Possibly at the expense of some local or global large cardinal assumptions.)

Or, is it the case that for any countable transitive model $M$, there is a $\Sigma^1_2$ (lightface) statement true in $V$ which fails in $M$?

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    $\begingroup$ If $V$ satisfies $V=L$, but $M$ is a model of, say Projective Determinacy, then there's really not that much hope for absoluteness. I think. $\endgroup$ – Asaf Karagila Sep 9 '18 at 20:17
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    $\begingroup$ You have several correctness results along these lines from large cardinal assumptions. See for instance section 7.2 in Steel's Handbook article. $\endgroup$ – Andrés E. Caicedo Sep 9 '18 at 21:15

First, it is consistent that there is a ctm but no $\Sigma^1_2$-correct ctm, for example if $V$ is the minimal model with a ctm. If there is a model of ZFC $M$ containing all the reals (e.g., if there is an inaccessible), then there is a projectively correct ctm: take the transitive collapse $H$ of a countable elementary substructure of $M$. Since reals don't move in the collapse, $H$ is projectively correct with real parameters. (By this argument there is a projectively correct model of ZFC if and only if there is a projectively correct ctm.)

As Andrés points out, one can also obtain correct models from determinacy hypotheses, though these hypotheses are much stronger than the existence of such models. E.g., if $\Pi^1_1$-determinacy holds then there is a $\Sigma^1_2$-correct ctm (e.g., $L_\kappa$ where $\kappa$ is the least Silver indiscernible).

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