论文标题
Varsovian型号$ω$
Varsovian models $ω$
论文作者
论文摘要
对于$ n <ω$，让$ n_n $为最低含量的适当的鼠标$ m $，使$ m \型号$“有序列$Δ_0<κ__0<κ__0<κ__0<\δ_{n-1} <κ___________________{n-1}同样，但是使用$ M \型号$，“有一个序列$λ$，它是伍丁红衣主教的限制和强烈的红衣主教的限制”。在适当的大型主要假设下，萨尔格森（Sargsyan）和辛德勒（Schindler）在“ Varsovian模型I”中介绍和分析了$ n_1 $的Varsovian模型，Sargsyan，Schindler和作者在“ Varsovian Models II”中介绍并分析了$ N_2 $的Varsovian Models II。假设$*$ - 转换与本文的p构建结构（尚未完成）常规集成在一起，我们将其扩展到$N_Ω$。我们在此假设下表明，$n_Ω$具有适当的类内部型号$ \ mathscr {v}_Ω$，它是一种完全触及的策略鼠标，具有$ω$ woodin cardinals，在其策略下关闭，而$ \ mathscr {v}_Ω$的宇宙是最终的通用hod hod，是$ n_的$ n_ $ n_ $ n_ $。我们还以相同的假设表明，$n_Ω$的核心模型$ k $（可以自然方式定义）是$n_Ω$的迭代，是$ \ mathscr {v}_Ω$的内部模型，并且在$ m $中是完全含糊的，并且在$ \ mathscr {v}_Ω_ $中是完全含糊的。
For $n<ω$, let $N_n$ be the minimal iterable proper class mouse $M$ such that $M\models$ "there are ordinals $δ_0<κ_0<\ldots<δ_{n-1}<κ_{n-1}$ such that each $δ_i$ is a Woodin cardinal and each $κ_i$ is a strong cardinal", and let $N_ω$ be likewise, but with $M\models$ "there is an ordinal $λ$ which is a limit of Woodin cardinals and a limit of strong cardinals". Under appropriate large cardinal hypotheses, Sargsyan and Schindler introduced and analysed in "Varsovian models I" the Varsovian model of $N_1$, and Sargsyan, Schindler and the author introduced and analysed in "Varsovian models II" the Varsovian model of $N_2$. We extend this to $N_ω$, assuming that $*$-translation integrates routinely with the P-constructions of this paper (the write-up of which is yet to be completed). We show, under this assumption, that $N_ω$ has a proper class inner model $\mathscr{V}_ω$ which is a fully iterable strategy mouse with $ω$ Woodin cardinals, closed under its strategy, and that the universe of $\mathscr{V}_ω$ is the eventual generic HOD, and the mantle, of $N_ω$. We also show, under the same assumption, that the core model $K$ of $N_ω$ (which can be defined in a natural manner) is an iterate of $N_ω$, is an inner model of $\mathscr{V}_ω$, and is fully iterable in $M$ and in $\mathscr{V}_ω$.
