Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations

TL;DR

This paper explores the deep connections between Martin's conjecture in recursion theory and countable Borel equivalence relations, highlighting a key result that arithmetic equivalence is universal among such relations.

Contribution

It provides an overview of work on Martin's conjecture and presents a significant unpublished result that arithmetic equivalence is a universal countable Borel equivalence relation.

Findings

Arithmetic equivalence is a universal countable Borel equivalence relation

Connections established between Martin's conjecture and descriptive set theory

Implications for the theory of universal countable Borel equivalence relations

Abstract

There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations. In this paper, we shall give an overview of some work that has been done on Martin's conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation. This theorem has interesting corollaries for the theory of universal countable Borel equivalence relations in general. We end with some open problems, and directions for future research.