Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.07623 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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.