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Ultrapowers of determinacy models as iteration trees on HOD

Gabriel Goldberg, Grigor Sargsyan, Benjamin Siskind

Abstract

In the 1990s, Steel and Woodin showed that under large cardinal hypotheses, the HOD of $L(\mathbb R)$ admits a fine-structural analysis. Although this theorem sheds light on various problems in descriptive set theory, the fine-structural representations of many fundamental objects of determinacy theory are still unknown. For example, Woodin asked whether the ultrapower of HOD by the closed unbounded filter on $ω_1$ is given by an iteration tree on HOD according to its fine-structural extender sequence and canonical iteration strategy. In this paper, we give a positive answer to Woodin's question, not only for the closed unbounded filter but for any ultrafilter on an ordinal. The key tool that enables the solution of Woodin's problem is a recent advance in inner model theory: the Steel--Schlutzenberg theory of normalizing iteration trees, which allows us to represent HOD and its ultrapowers as normal iterates of a single countable mouse. Despite our results, the precise structure of the iteration trees that lead from HOD into its ultrapowers remains a mystery.

Ultrapowers of determinacy models as iteration trees on HOD

Abstract

In the 1990s, Steel and Woodin showed that under large cardinal hypotheses, the HOD of admits a fine-structural analysis. Although this theorem sheds light on various problems in descriptive set theory, the fine-structural representations of many fundamental objects of determinacy theory are still unknown. For example, Woodin asked whether the ultrapower of HOD by the closed unbounded filter on is given by an iteration tree on HOD according to its fine-structural extender sequence and canonical iteration strategy. In this paper, we give a positive answer to Woodin's question, not only for the closed unbounded filter but for any ultrafilter on an ordinal. The key tool that enables the solution of Woodin's problem is a recent advance in inner model theory: the Steel--Schlutzenberg theory of normalizing iteration trees, which allows us to represent HOD and its ultrapowers as normal iterates of a single countable mouse. Despite our results, the precise structure of the iteration trees that lead from HOD into its ultrapowers remains a mystery.
Paper Structure (7 sections, 24 theorems, 3 equations)

This paper contains 7 sections, 24 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The HOD analysis
  4. Full normalization
  5. Main theorems
  6. $\textnormal{AD}_\mathbb{R}+V=L(P(\mathbb{R}))+\textnormal{HPC}$
  7. $V = L(\mathbb{R})$

Key Result

Lemma 2.3

Assume $\textnormal{AD}^+$. Let $(P, \Sigma)$ be a least branch hod pair with scope $\textnormal{HC}$. Then $\Sigma^+$ restricts to a total iteration strategy for normal trees of length less than $\Theta$. That is, whenever $\mathcal{T}$ is by $\Sigma^+$ and has limit length less than $\Theta$, $\Si

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Steel, mousepairs
  • Theorem 2.4: Steel-Woodin, hodas
  • Theorem 2.5: Steel, mousepairs
  • Definition 2.6
  • Theorem 2.7: Steel, mousepairs
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 2.10: Steel steel-book, Schlutzenberg fullnormalization
  • ...and 27 more