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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2304.07623 |
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| _version_ | 1866912326998818816 |
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| author | Müller, Sandra Sargsyan, Grigor |
| author_facet | Müller, Sandra Sargsyan, Grigor |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07623 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards a generic absoluteness theorem for Chang models Müller, Sandra Sargsyan, Grigor Logic 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. |
| title | Towards a generic absoluteness theorem for Chang models |
| topic | Logic |
| url | https://arxiv.org/abs/2304.07623 |