How many Turing degrees are there?

TL;DR

This paper surveys the hierarchy of countable Borel equivalence relations, focusing on their complexity and classification within recursion theory, including Turing degrees and related equivalence relations.

Contribution

It provides a comprehensive overview of the complexity hierarchy of countable Borel equivalence relations in recursion theory, highlighting open problems and conjectures.

Findings

Turing equivalence's position in the Borel hierarchy

Connections between recursion theory and Borel reducibility

Open problems in classifying countable Borel equivalence relations

Abstract

A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence relations imposed by the notion of Borel reducibility. We will survey results and conjectures concerning the problem of identifying the place in this hierarchy of these equivalence relations from recursion theory and also discuss some of their implications.