Guardat en:
| Autor principal: | |
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| Format: | Preprint |
| Publicat: |
2020
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| Matèries: | |
| Accés en línia: | https://arxiv.org/abs/2012.07185 |
| Etiquetes: |
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Taula de continguts:
- Let $M$ be a tame mouse modelling ZFC. We show that $M$ satisfies "$V=\mathrm{HOD}_x$ for some real $x$", and that the restriction $\mathbb{E}\upharpoonright[ω_1^M,\mathrm{OR}^M)$ of the extender sequence $\mathbb{E}^M$ of $M$ to indices above $ω_1^M$ is definable without parameters over the universe of $M$. We show that $M$ has universe $\mathrm{HOD}^M[X]$, where $X=M|ω_1^M$ is the initial segment of $M$ of height $ω_1^M$ (including $\mathbb{E}^M\upharpoonrightω_1^M$), and that $\mathrm{HOD}^M$ is the universe of a premouse over some $t\subseteqω_2^M$. We also show that $M$ has no proper grounds via strategically $σ$-closed forcings. We then extend some of these results partially to non-tame mice, including a proof that many natural $φ$-minimal mice model "$V=\mathrm{HOD}$", assuming a certain fine structural hypothesis whose proof has almost been given elsewhere.