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On a question of Slaman and Steel

Adam Day, Andrew Marks

TL;DR

This work links Slaman and Steel's question about representing $\equiv_T$ as an increasing union of Borel relations to the broader notion of hyper-Borel-finiteness and to Martin's conjecture. It argues that a positive answer would entail striking consequences, including a universal countable Borel relation that is not uniformly universal and the existence of a $(\equiv_T,\equiv_m)$-invariant function that is not uniformly invariant on any pointed perfect set. The authors provide the first nonuniform construction (Theorem m_construction) of such degree-structure invariants, and develop strengthened Kuratowski–Mycielski-type tools to support these constructions. Overall, the paper reveals deep connections between hyperfiniteness, invariant function theory, and Martin-type conjectures within the study of countable Borel equivalence relations.

Abstract

We consider an old question of Slaman and Steel: whether Turing equivalence is an increasing union of Borel equivalence relations none of which contain a uniformly computable infinite sequence. We show this question is deeply connected to problems surrounding Martin's conjecture, and also in countable Borel equivalence relations. In particular, if Slaman and Steel's question has a positive answer, it implies there is a universal countable Borel equivalence which is not uniformly universal, and that there is a $(\equiv_T,\equiv_m)$-invariant function which is not uniformly invariant on any pointed perfect set.

On a question of Slaman and Steel

TL;DR

This work links Slaman and Steel's question about representing as an increasing union of Borel relations to the broader notion of hyper-Borel-finiteness and to Martin's conjecture. It argues that a positive answer would entail striking consequences, including a universal countable Borel relation that is not uniformly universal and the existence of a -invariant function that is not uniformly invariant on any pointed perfect set. The authors provide the first nonuniform construction (Theorem m_construction) of such degree-structure invariants, and develop strengthened Kuratowski–Mycielski-type tools to support these constructions. Overall, the paper reveals deep connections between hyperfiniteness, invariant function theory, and Martin-type conjectures within the study of countable Borel equivalence relations.

Abstract

We consider an old question of Slaman and Steel: whether Turing equivalence is an increasing union of Borel equivalence relations none of which contain a uniformly computable infinite sequence. We show this question is deeply connected to problems surrounding Martin's conjecture, and also in countable Borel equivalence relations. In particular, if Slaman and Steel's question has a positive answer, it implies there is a universal countable Borel equivalence which is not uniformly universal, and that there is a -invariant function which is not uniformly invariant on any pointed perfect set.

Paper Structure

This paper contains 7 sections, 13 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Versions of Martin's conjecture for invariant functions from Turing to many-one degrees
  4. Slaman and Steel's Question
  5. Strengthenings of the Kuratowski-Mycielski theorem
  6. A nonuniform construction
  7. Open questions

Key Result

Proposition 2.4

Conjecture conj:strong_steel_for_m implies Steel's conjecture, SS.

Theorems & Definitions (36)

  • Conjecture 2.1: Martin's conjecture for $(\equiv_T,\equiv_m)$-invariant functions
  • Conjecture 2.2: Steel's conjecture for $(\equiv_T,\equiv_m)$-invariant functions
  • Conjecture 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5: Kihara and Montalbán, private communication
  • proof
  • Definition 3.1: SS
  • Theorem 3.5
  • proof
  • ...and 26 more