Ordinal Definability and Combinatorics of Equivalence Relations

2017/11/12 by William Chan, Chan, William
Computer Science · Mathematics · #Advanced Topology and Set Theory #Coding theory and cryptography #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1711.04353

openalex publication_date 2017/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assume \mathsfZF + AD⁺ + V = L(\mathscrP(ℝ)). Let E be a \mathbfΣ¹₁ equivalence relation coded in HOD. E has an ordinal definable equivalence class without any ordinal definable elements if and only if HOD \models E is unpinned. \mathsfZF + AD⁺ + V = L(\mathscrP(ℝ)) proves E-class section uniformization when E is a \mathbfΣ¹₁ equivalence relation on ℝ which is pinned in every transitive model of ZFC containing the real which codes E: Suppose R is a relation on ℝ such that each section R_x = \y : (x,y) ∈ R\ is an E-class, then there is a function f : ℝ → ℝ such that for all x ∈ ℝ, R(x,f(x)). ZF + AD proves that ℝ × κ is Jónsson whenever κ is an ordinal: For every function f : [ℝ × κ]^

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