Projective Determinacy from long Chang's Conjecture

TL;DR

This paper demonstrates that a specific form of Chang's Conjecture combined with GCH implies projective determinacy, using a novel covering argument for mouse reflection.

Contribution

It introduces a new approach using a variant covering argument to connect Chang's Conjecture with projective determinacy.

Findings

Chang's Conjecture plus GCH implies projective determinacy

A novel covering argument for mouse reflection is developed

Potential applicability to other forms of Chang's Conjecture

Abstract

Consider the property $(\aleph_{\omega + 1},\aleph_{\omega + 2},\ldots) \twoheadrightarrow (\aleph_1,\aleph_2,\ldots)$. Here we will show that this property with the addition of the General Continuum Hypothesis implies projective determinacy. Of particular interest here is the use of a variant covering argument to prove limited instances of mouse reflection. We believe that this approach could find use for other forms of Chang's Conjecture as well.