The definability of the extender sequence $\mathbb{E}$ from $\mathbb{E}{\upharpoonright}\aleph_1$ in $L[\mathbb{E}]$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866910907651588096 |
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| author | Schlutzenberg, Farmer |
| author_facet | Schlutzenberg, Farmer |
| contents | Let $M$ be a short extender mouse. We prove that if $E\in M$ and $M$ satisfies "$E$ is a countably complete short extender whose support is a cardinal $θ$ and $\mathcal{H}_θ\subseteq\mathrm{Ult}(V,E)$", then $E$ is in the extender sequence $\mathbb{E}^M$ of $M$. We also prove other related facts, and use them to establish that if $κ$ is an uncountable cardinal of $M$ and $κ^{+M}$ exists in $M$ then $(\mathcal{H}_{κ^+})^M$ satisfies the Axiom of Global Choice.
We prove that if $M$ satisfies the Power Set Axiom then $\mathbb{E}^M$ is definable over the universe of $M$ from the parameter $X=\mathbb{E}^M\upharpoonright\aleph_1^M$, and $M$ satisfies "every set is $\mathrm{OD}_{\{X\}}$". We also prove various local versions of this fact in which $M$ has a largest cardinal, and a version for generic extensions of $M$.
As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models "$V=\mathrm{HOD}$". This adapts to many other similar examples.
We also describe a simplified approach to Mitchell-Steel fine structure, which does away with the parameters $u_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_00276 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The definability of the extender sequence $\mathbb{E}$ from $\mathbb{E}{\upharpoonright}\aleph_1$ in $L[\mathbb{E}]$ Schlutzenberg, Farmer Logic 03E45, 03E55 Let $M$ be a short extender mouse. We prove that if $E\in M$ and $M$ satisfies "$E$ is a countably complete short extender whose support is a cardinal $θ$ and $\mathcal{H}_θ\subseteq\mathrm{Ult}(V,E)$", then $E$ is in the extender sequence $\mathbb{E}^M$ of $M$. We also prove other related facts, and use them to establish that if $κ$ is an uncountable cardinal of $M$ and $κ^{+M}$ exists in $M$ then $(\mathcal{H}_{κ^+})^M$ satisfies the Axiom of Global Choice. We prove that if $M$ satisfies the Power Set Axiom then $\mathbb{E}^M$ is definable over the universe of $M$ from the parameter $X=\mathbb{E}^M\upharpoonright\aleph_1^M$, and $M$ satisfies "every set is $\mathrm{OD}_{\{X\}}$". We also prove various local versions of this fact in which $M$ has a largest cardinal, and a version for generic extensions of $M$. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models "$V=\mathrm{HOD}$". This adapts to many other similar examples. We also describe a simplified approach to Mitchell-Steel fine structure, which does away with the parameters $u_n$. |
| title | The definability of the extender sequence $\mathbb{E}$ from $\mathbb{E}{\upharpoonright}\aleph_1$ in $L[\mathbb{E}]$ |
| topic | Logic 03E45, 03E55 |
| url | https://arxiv.org/abs/1906.00276 |