Towards a generic absoluteness theorem for Chang models

Abstract

Let Γ∞ be the set of all universally Baire sets of reals. Inspired by the work done in [22] and [21], we introduce a new technique for establishing generic absoluteness results for models containing Γ∞. Our main technical tool is an iteration that realizes Γ∞ 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⊆Col(ω,2^2^κ) is V-generic and h is V[g]-generic for some poset P∈V[g], there is an elementary embedding j:V→M such that j(κ)=ω_1^{V[g⁎h]} and L(Γ∞,R) as computed in V[g⁎h] is a derived model of M at j(κ). Here j is obtained by iteratively taking ultrapowers of V by extenders with critical point κ and its images. As a corollary we obtain that 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 AD_R+“Θ is a regular cardinal”. Inspired by core model induction, we introduce the definable powerset A∞ of Γ∞ and use our derived model representation mentioned above to show that the theory of L(A∞) cannot be changed by forcing (see Theorem 1.19). Working in a different direction, we also show that the theory of L(Γ∞,R)[C], where C is the club filter on ℘_{ω_1}(Γ∞), cannot be changed by forcing (see Theorem 1.30). Proving the two aforementioned results is the first step towards showing that the theory of L(Ord^ω,Γ∞,R)([μ_α : α∈Ord]), where μ_α is the club filter on ℘_{ω_1}(α), cannot be changed by forcing.