Towards a generic absoluteness theorem for Chang models

Abstract

Let $\Gamma^\infty$ be the set of all universally Baire sets of reals. Inspired by recent work of the second author and Nam Trang, we introduce a new technique for establishing generic absoluteness results for models containing $\Gamma^\infty$. Our main technical tool is an iteration that realizes $\Gamma^\infty$ as the sets of reals in a derived model of some iterate of $V$. We show, from a supercompact cardinal $\kappa$ and a proper class of Woodin cardinals, that whenever $g \subseteq Col(\omega, 2^{2^\kappa})$ is $V$-generic and $h$ is $V[g]$-generic for some poset $\mathbb{P}\in V[g]$, there is an elementary embedding $j: V\rightarrow M$ such that $j(\kappa)=\omega_1^{V[g*h]}$ and $L(\Gamma^\infty, \mathbb{R})$ as computed in $V[g*h]$ is a derived model of $M$ at $j(\kappa)$. As a corollary we obtain that $\mathsf{Sealing}$ holds in $V[g]$, which was previously demonstrated by Woodin using the stationary tower forcing. Also, using a theorem of Woodin, we conclude that the derived model of $V$ at $\kappa$ satisfies $\mathsf{AD}_{\mathbb{R}}+``\Theta$ is a regular cardinal". Inspired by core model induction, we introduce the definable powerset $\mathcal{A}^\infty$ of $\Gamma^\infty$ and use our derived model representation mentioned above to show that the theory of $L(\mathcal{A}^\infty)$ cannot be changed by forcing. Working in a different direction, we also show that the theory of $L(\Gamma^\infty, \mathbb{R})[\mathcal{C}]$, where $\mathcal{C}$ is the club filter on $\wp_{\omega_1}(\Gamma^\infty)$, cannot be changed by forcing. Proving the two aforementioned results is the first step towards showing that the theory of $L(Ord^\omega, \Gamma^\infty, \mathbb{R})([\mu_\alpha: \alpha\in Ord])$, where $\mu_\alpha$ is the club filter on $\wp_{\omega_1}(\alpha)$, cannot be changed by forcing.