Determinacy and Large Cardinals

TL;DR

This paper explores the deep connections between determinacy hypotheses in set theory and the existence of inner models with large cardinals, highlighting recent advances in the field.

Contribution

It reviews key concepts and recent progress linking determinacy hypotheses with the existence of mice containing large cardinals.

Findings

Existence of mice with Woodin cardinals suffices for projective determinacy.

Recent progress improves understanding of the equivalence between determinacy hypotheses and large cardinal models.

Connections between inner models and determinacy hypotheses are further clarified.

Abstract

The study of inner models was initiated by G\"odel's analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin's proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel's proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later Neeman improved the result in the projective hierarchy by showing that in fact the existence of a countable iterable model, a mouse, with Woodin cardinals and a top measure suffices to prove determinacy in the projective hierarchy. This opened up the possibility for an optimal result stating the equivalence between local determinacy hypotheses and the existence of mice in the projective hierarchy. This article outlines the main concepts and results connecting determinacy hypotheses with the existence of mice with large cardinals as well as recent progress in the area.