Ideals and Strong Axioms of Determinacy

TL;DR

This paper establishes the equiconsistency between two complex set-theoretic theories involving ideals, determinacy, and large cardinals, resolving longstanding conjectures and core model induction problems.

Contribution

It proves that the minimal model of a theory with ideals and determinacy exists if and only if a related theory with large cardinals is consistent, linking two major frameworks in set theory.

Findings

Proves the equiconsistency of theories T and S.

Shows that T implies the existence of the minimal model of S.

Resolves a longstanding conjecture of Woodin.

Abstract

We show that the following two theories are equiconsistent: (T) ZFC, CH and "There is a dense ideal on the first uncountable cardinal such that if j is the generic embedding associated with it then its restriction on ordinals is independent of the generic object is". (S) ZF, ADR and "Theta is a regular cardinal." The main result of this paper is that T implies that the minimal model of S exists. Woodin, in unpublished work, showed that the consistency of S implies the consistency of T. We will also give a proof of this result, which, together with our main theorem, establishes the equiconsistency of T and S. Our main result partially resolves a well-known conjecture of Woodin, and completely solves one of the main Core Model Induction problems dating back to 90s.