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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.08607 |
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| _version_ | 1866916612421976064 |
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| author | Gappo, Takehiko Müller, Sandra Sargsyan, Grigor |
| author_facet | Gappo, Takehiko Müller, Sandra Sargsyan, Grigor |
| contents | Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal $δ$ that is both a limit of Woodin cardinals and a limit of ${<}δ$-strong cardinals. The existence of such a hod mouse is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. We argue that our Chang-type model satisfies $\mathsf{AD}_{\mathbb{R}} + Θ$ is regular + $ω_1$ is ${<}δ_{\infty}$-supercompact for some regular cardinal $δ_{\infty}>Θ$. This complements Woodin's generalized Chang model, which satisfies $\mathsf{AD}_{\mathbb{R}}+ω_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08607 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Chang models over derived models with supercompact measures Gappo, Takehiko Müller, Sandra Sargsyan, Grigor Logic Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal $δ$ that is both a limit of Woodin cardinals and a limit of ${<}δ$-strong cardinals. The existence of such a hod mouse is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. We argue that our Chang-type model satisfies $\mathsf{AD}_{\mathbb{R}} + Θ$ is regular + $ω_1$ is ${<}δ_{\infty}$-supercompact for some regular cardinal $δ_{\infty}>Θ$. This complements Woodin's generalized Chang model, which satisfies $\mathsf{AD}_{\mathbb{R}}+ω_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals. |
| title | Chang models over derived models with supercompact measures |
| topic | Logic |
| url | https://arxiv.org/abs/2307.08607 |