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ORDINAL DEFINABILITY IN $L[\mathbb {E}]$
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PRE
AI
DOI：10.1017/jsl.2025.10099.png)
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Abstract
En 中文
Let M be a tame mouse modelling $\mathrm {ZFC}$ . We show that M satisfies $V=\mathrm {HOD}_x$ for some real x, and that the restriction $\mathbb {E}<^>M\!\upharpoonright \![\omega _1<^>M,\mathrm {OR}<^>M)$ of the extender sequence $\mathbb {E}<^>M$ of M to indices above $\omega _1<^>M$ is definable without parameters over the universe of M. We show that M has universe $\mathrm {HOD}<^>M[X]$ , where $X=M|\omega _1<^>M$ is the initial segment of M of height $\omega _1<^>M$ (including $\mathbb {E}<^>M\!\upharpoonright \!\omega _1<^>M$ ), and that $\mathrm {HOD}<^>M$ is the universe of a premouse over some $t\subseteq \omega _2<^>M$ . We also show that M has no proper grounds via strategically $\sigma $ -closed forcings. We then extend some of these results partially to non-tame mice, including a proof that many natural $\varphi $ -minimal mice model $V=\mathrm {HOD}$ , assuming a certain fine structural hypothesis whose proof was almost established in Closson [1], and has since been completed in the preprint [14].
Keywords:
inner model
mouse
definability
HOD
self-iterability