Towards a generic absoluteness theorem for Chang models
Sandra Müller, Grigor Sargsyan
TL;DR
The paper advances generic absoluteness for Chang-type models by representing L(Γ^∞, R) as a derived model arising from a DM-sequence built via one-step genericity constructions and stabilization under large-cardinal collapses. It extends Woodin's Sealing framework to the universally Baire setting, showing that forcing can preserve theories of key definable powersets such as L(Γ^∞, R) and L(P_{uB}(Γ^∞), R), and even L(Ord^ω, Γ^∞, R)[C^∞], under appropriate hypotheses (e.g., a supercompact κ and a class of Woodin cardinals). The paper develops the UB-Capturing Principle, the one-step construction, and a derived-model representation to prove Sealing and Generically Correct Sealing, with corollaries including Theta-regularity and robustness under forcing. These results contribute to a program aiming to render the theory of canonical determinations over Γ^∞ and related clubs and uB-structures generic-absolute, potentially reducing aspects of MM^{++} over determinacy to canonical determinacy-based inner models. Overall, the work provides a coherent framework to realize and preserve complex definable powersets through derived-model techniques in a richly large-cardinal environment, advancing our understanding of forcing-invariant theories for Γ^∞ and related objects.
Abstract
Let $Γ^\infty$ 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 $Γ^\infty$. Our main technical tool is an iteration that realizes $Γ^\infty$ as the sets of reals in a derived model of some iterate of $V$. We show, from a supercompact cardinal $κ$ and a proper class of Woodin cardinals, that whenever $g \subseteq Col(ω, 2^{2^κ})$ 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 $j(κ)=ω_1^{V[g*h]}$ and $L(Γ^\infty, \mathbb{R})$ as computed in $V[g*h]$ is a derived model of $M$ at $j(κ)$. 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 $κ$ satisfies $\mathsf{AD}_{\mathbb{R}}+``Θ$ is a regular cardinal". Inspired by core model induction, we introduce the definable powerset $\mathcal{A}^\infty$ of $Γ^\infty$ 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 $L(Γ^\infty, \mathbb{R})[\mathcal{C}]$, where $\mathcal{C}$ is the club filter on $\wp_{ω_1}(Γ^\infty)$, cannot be changed by forcing. Proving the two aforementioned results is the first step towards showing that the theory of $L(Ord^ω, Γ^\infty, \mathbb{R})([μ_α: α\in Ord])$, where $μ_α$ is the club filter on $\wp_{ω_1}(α)$, cannot be changed by forcing.