Ideals and Strong Axioms of Determinacy
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 Woodin, in unpublished work, showed that the consistency of S implies the consistency of T. We will also give a proof of this result, which, together with our main theorem,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.
Keywords
Cite
@article{arxiv.2111.06220,
title = {Ideals and Strong Axioms of Determinacy},
author = {Dominik Adolf and Grigor Sargsyan and Nam Trang and Trevor Wilson and Martin Zeman},
journal= {arXiv preprint arXiv:2111.06220},
year = {2022}
}
Comments
arXiv admin note: text overlap with arXiv:1608.05726