When an Equivalence Relation with All Borel Classes will be Borel Somewhere?

2016/08/17 by William Chan, Menachem Magidor, Chan, William +1
Mathematics · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1608.04913

Abstract

In ZFC, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation E ∈ L(ℝ) on ℝ with all \mathbfΔ₁¹ classes and every σ-ideal I on ℝ so that the associated forcing ℙ_I of I⁺ \mathbfΔ₁¹ subsets is proper, there exists some I⁺ \mathbfΔ₁¹ set C so that E \upharpoonright C is a \mathbfΔ₁¹ equivalence relation. In ZF + DC + AD_ℝ + V = L(\mathscrP(ℝ)), for every equivalence relation E on ℝ with all \mathbfΔ₁¹ classes and every σ-ideal I on ℝ so that the associated forcing ℙ_I is proper, there is some I⁺ \mathbfΔ₁¹ set C so that E \upharpoonright C is a \mathbfΔ₁¹ equivalence relation.

Related

• Equivalence Relations Which Are Borel Somewhere
• The Effective Theory of Borel Equivalence Relations
• The Countable Admissible Ordinal Equivalence Relation
• On large cardinals and generalized Baire spaces
• Borel Canonization of Analytic Sets with Borel Sections