Varsovian models II

Abstract

Assume the existence of sufficent large cardinals. Let $M_{\mathrm{sw}n}$ be the minimal iterable proper class $L[E]$ model satisfying "there are $δ_0<κ_0<\ldots<δ_{n-1}<κ_{n-1}$ such that the $δ_i$ are Woodin cardinals and the $κ_i$ are strong cardinals". Let $M=M_{\mathrm{sw}2}$. We identify an inner model $\mathscr{V}_2^M$ of $M$, which is a proper class model satisfying "there are 2 Woodin cardinals", and is iterable both in $V$ and in $M$, and closed under its own iteration strategy. The construction also yields significant information about the extent to which $M$ knows its own iteration strategy. We characterize the universe of $\mathscr{V}_2^M$ as the mantle and the least ground of $M$, and as $\mathrm{HOD}^{M[G]}$ for $G\subseteq\mathrm{Coll}(ω,λ)$ being $M$-generic with $λ$ sufficiently large. These results correspond to facts already known for $M_{\mathrm{sw}1}$, and the proofs are an elaboration of those, but there are substantial new issues and new methods used to handle them.