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Often, large cardinals imply that many definable inner models and transitive sets have theories which are absolute under forcing. For example, assuming a proper class of Woodin cardinals, the theory of $L(\mathbb{R})$ is absolute under forcing in the following sense: if $V[G]$ is a set-generic extension of $V$, then $H(\omega_1)^V\preccurlyeq H(\omega_1)^{V[G]}$. See this paper of Bagaria for more facts about forcing absoluteness.

I'm curious about a possible absoluteness principle for a much larger model. The Chang model, $C$, is the smallest model of set theory containing all the ordinals, and closed under countable sequences. My question is:

What is the large cardinal strength of the statement, "The lightface theory of the Chang model is forcing absolute?"

That is, for every sentence $\varphi$ in the language of set theory without parameters and every set-generic extension $V[G]$, we have $C^V\models\varphi\iff C^{V[G]}\models\varphi$.

It is not at all obvious to me that this is consistent - in fact, right now I suspect it isn't - but I don't see how to get a contradiction from it.

(We can also ask the same question involving some parameters - say, real parameters - of sufficiently low complexity that the obvious way of building a contradiction doesn't work. However, I don't want to get greedy - for now I'm interested in specifically the purely lightface version.)

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  • $\begingroup$ Corollary 3.1.7 in Larson's Stationary Tower notes gives that if $\delta$ is a Woodin limit of Woodin cardinals, then no forcing in $V_\delta$ can change the theory of the Chang model, even with real parameters. Corollary 3.1.10 establishes the same thing when $\delta$ is strongly compact. Am I missing something, or is this not sufficient? $\endgroup$ – Cameron Zwarich Dec 7 '16 at 0:29
  • $\begingroup$ @CameronZwarich Nope, that's exactly what I was looking for - thanks a ton! If you post that as an answer, I'll accept it. $\endgroup$ – Noah Schweber Dec 7 '16 at 0:39
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Corollary 3.1.7 in Larson's Stationary Tower notes states that if $\delta$ is a Woodin limit of Woodin cardinals, then no forcing in $V_\delta$ can change the theory of the Chang model, even with real parameters. Corollary 3.1.10 states the same thing when $\delta$ is strongly compact.

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  • $\begingroup$ A silly question just occurred to me (I don't have Larson's book on hand): is he assuming the $\Omega$-conjecture or similar, or is this an outright proof? $\endgroup$ – Noah Schweber Dec 7 '16 at 0:41
  • $\begingroup$ It's an outright proof. $\endgroup$ – Cameron Zwarich Dec 7 '16 at 0:53
  • $\begingroup$ In case you look it up later, I should note that there are some errata for the proof of the second result, indicating that it's not actually a corollary of Theorem 3.1.9 but is still true. $\endgroup$ – Cameron Zwarich Dec 7 '16 at 0:59

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