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As long as $T_\alpha$ is uniformly definable say in $V_{\alpha+\omega}$, $V_\kappa$ is a model of $T_\kappa$ for $\kappa$ inaccessible. So you can't get "there is an inaccessible" that way.

If I understand the question correctly, then Woodin's Σ2 resurrection theorem gives examples. Assume there is a proper class of Woodin cardinals, and let φ be a Σ2 statement which holds in V. Let P be …

There is a useful notion of "weak Woodinness" according to which every uncountable regular cardinal is weakly Woodin :) See https://ivv5hpp.uni-muenster.de/u/rds/fabiana_ralf.pdf

Re a proof of this result: I once wrote up an elementary proof of this which avoids the Martin-Solovay tree: See Theorem 1 of https://ivv5hpp.uni-muenster.de/u/rds/talks-barcelona.dvi

$SCH$ follows from $MM$ (Foreman-Magidor-Shelah), in fact from $PFA$ (Viale). One reason why one might not want to believe $MM$ is that there is another ``maximality principle'' which also implies $2^ …

Nice question! But which aged set theorist drinks prominent Scotch? The following does not answer this question either. Let $T \in L$ be your favorite Suslin tree of $L$. Consider the $L({\mathbb R}) …

If κ is weakly compact in VCol(κ,κ+), then there is a non-domestic premouse, see Theorem 0.4 of https://ivv5hpp.uni-muenster.de/u/rds/stacking_mice.pdf

To show this in L(R) (or any well understood determinacy model) one apparently has to run a core model induction as in section 6.2 of https://ivv5hpp.uni-muenster.de/u/rds/core_model_induction.pdf w …

There is rumor that Silver's efforts started as an attempt to come up with a flawless argument showing the main "theorem" in Jensen's "A modest remark." That main "theorem" says that in ZF, there is n …