Varsovian models II

TL;DR

This paper constructs an inner model with two Woodin cardinals within a larger model assuming large cardinals, and explores its properties and relation to the universe and ground models.

Contribution

It identifies a new inner model with two Woodin cardinals that is iterable and closed under its iteration strategy, extending previous results for one Woodin cardinal.

Findings

The inner model $ extstyle ext{ extgoth V}_2^M$ has two Woodin cardinals.

$ ext{ extgoth V}_2^M$ is the mantle and least ground of $M$.

$ ext{ extgoth V}_2^M$ equals $ ext{HOD}^{M[G]}$ for certain generic extensions.

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 $\delta_0<\kappa_0<\ldots<\delta_{n-1}<\kappa_{n-1}$ such that the $\delta_i$ are Woodin cardinals and the $\kappa_i$ 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 $G\subseteq\mathrm{Coll}(\omega,\lambda)$ being $M$-generic with $\lambda$ 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.