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Varsovian models II

Grigor Sargsyan, Ralf Schindler, Farmer Schlutzenberg

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.

Varsovian models II

Abstract

Assume the existence of sufficent large cardinals. Let be the minimal iterable proper class model satisfying "there are such that the are Woodin cardinals and the are strong cardinals". Let . We identify an inner model of , which is a proper class model satisfying "there are 2 Woodin cardinals", and is iterable both in and in , and closed under its own iteration strategy. The construction also yields significant information about the extent to which knows its own iteration strategy. We characterize the universe of as the mantle and the least ground of , and as for being -generic with sufficiently large. These results correspond to facts already known for , and the proofs are an elaboration of those, but there are substantial new issues and new methods used to handle them.

Paper Structure

This paper contains 45 sections, 96 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Ground generation
  4. The model $M_{\mathrm{swsw}}$
  5. The first Varsovian model $\mathscr{V}_1$
  6. The models for the system
  7. The short tree strategy $\Sigma_\mathrm{sh}$ for $M$
  8. The first direct limit system
  9. The external direct limit system $\mathscr{D}^{\mathrm{ext}}$
  10. The internal direct limit system $\mathscr{D}$
  11. The first Varsovian model as $\mathcal{M}_\infty[*]$
  12. $\mathrm{HOD}^{M[G]}_{\mathscr{E}}$
  13. Uniform grounds
  14. The structure $\mathcal{Q}$
  15. The ${\kappa_0}$-mantle of $M$
  16. ...and 30 more sections

Key Result

Lemma 2

For each $s\in[\mathrm{OR}]^{<\omega}\backslash\{\emptyset\}$, there is $p\in d$ such that $s$ is $p$-stable.

Theorems & Definitions (288)

  • Remark 1
  • Definition 2.1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 278 more