Ordinal definability in $L[\mathbb{E}]$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918446164344832 |
|---|---|
| author | Schlutzenberg, Farmer |
| author_facet | Schlutzenberg, Farmer |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_07185 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Ordinal definability in $L[\mathbb{E}]$ Schlutzenberg, Farmer Logic 03E45, 03E55 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. |
| title | Ordinal definability in $L[\mathbb{E}]$ |
| topic | Logic 03E45, 03E55 |
| url | https://arxiv.org/abs/2012.07185 |