Ideals and Strong Axioms of Determinacy

Dominik Adolf; Grigor Sargsyan; Nam Trang; Trevor Wilson; Martin Zeman

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arXiv · 2111.06220

Ideals and Strong Axioms of Determinacy

Dominik Adolf·Grigor Sargsyan·Nam Trang·Trevor Wilson·Martin Zeman

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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 Woodin, in unpublished work, showed that the consistency of S implies the consistency of T. We will also give a proof of this result, which, together with our main theorem,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.

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BibTeX RIS

Dominik Adolf, Grigor Sargsyan, Nam Trang, Trevor Wilson, Martin Zeman. 2021-11-11. Ideals and Strong Axioms of Determinacy. https://arxiv.org/abs/2111.06220

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