How Many Turing Degrees are There ?

A Borel equivalence relation on a Polish space is 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. The obvious answer to the question of the title is: continuum many. There is however a different way of looking at this question, which leads to some very interesting open problems in the interface of recursion theory and descriptive set theory. Our goal in this paper is to explain the context in which this and related problems can be formulated, i.e., the theory of Borel equivalence relations, and survey some of the progress to date. 1. Formulation of the problem We denote by =r the Turing equivalence relation on P(N) = {X : X <:;; N}, which we identify with 2N, viewing sets as characteristic functions. (We use the standard set-theoretic convention that n = {0, 1, ... , n 1} for all natural numbers n.) Then =r is a Borel (in fact~~) equivalence relation on 2N. We denote by V the quotient space 2N /(=r), i.e., the set of Turing degrees. Now consider general Borel equivalence relations on 2N or even arbitrary Polish (separable completely metrizable) spaces. We measure their complexity by studying the following partial (pre )order of Borel reducibility: if E, F are Borel equivalence relations on X, Y respectively, then a Borel reduction of E into F is a Borel map f : X ~ Y such that xEy -<====} f(x)Ff(y). If such an f exists we say that E is Borel reducible to F and denote this by E ~B F. Let also E "'B F -<====} E ~B F & F ~BE 1991 Mathematics Subject Classification. Primary 03D30, 03E15; Secondary 04A15, 54H05. The first author was partially supported by NSF Grant DMS 9158092. The second author was partially supported by NSF Grant DMS 9619880. © 2000 American 1\ilathemat ical Society