Ultrapowers of determinacy models as iteration trees on HOD

TL;DR

This paper proves that ultrapowers of HOD by any ultrafilter on an ordinal can be represented as iteration trees on HOD, advancing the understanding of inner model theory and determinacy models.

Contribution

It provides a positive answer to Woodin's question, showing ultrapowers of HOD are given by iteration trees, using the Steel--Schlutzenberg theory of normalizing iteration trees.

Findings

Ultrapowers of HOD can be represented as iteration trees.

The Steel--Schlutzenberg theory enables this representation.

The structure of iteration trees from HOD remains an open question.

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 $\omega_1$ 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.