论文标题
$ l [\ mathbb {e}] $中的序数确定性
Ordinal definability in $L[\mathbb{E}]$
论文作者
论文摘要
令$ m $为驯服的鼠标建模ZFC。我们表明，$ m $满足“ $ v = \ mathrm {hod} _x $ for Some $ x $”，并且限制$ \ Mathbb {e} \ upharpoonright [ω_1^m，\ mathrm {or}^m）可以在$ m $的宇宙上定义没有参数。我们表明，$ m $具有宇宙$ \ mathrm {hod}^m [x] $，其中$ x = m |ω_1^m $是$ m $高度的初始部分$ $ m $ of $ω_1^m $（包括$ \ mathbb {e}^m \ upharpoonRightω_1^m $），以及$ \ is的$ $ \ $ \ mather} $ t \subseteqΩ_2^m $。我们还表明，$ m $没有通过战略上$σ$ claus的强迫措施具有适当的理由。 然后，我们将其中的一些结果部分扩展到非焦化小鼠，包括许多天然的$φ$ - 最小小鼠模型“ $ v = \ mathrm {hod} $”，假设其几乎在其他地方给出了证明的证据。
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.
