Is the Turing Jump Unique? Martin’s Conjecture, and Countable Borel Equivalence Relations

In 1936, Alan Turing wrote a remarkable paper giving a negative answer to Hilbert’s Entscheidungsproblem [29]. Restated with modern terminology and in its relativized form, Turing showed that given any infinite binary sequence x ∈ 2ω, the set x′ of Turing machines that halt relative to x is not computable from x. This function x 7→ x′ is now known as the Turing jump, and it has played a singularly important role in the development of recursion theory, providing a canonical operator for increasing complexity in the Turing degrees. In this essay, we shall present an overview of developments which have lead to a line of research with the potential for precisely explaining the central role that the Turing jump plays in recursion theory, and more generally in the theory of definability in mathematics. The centerpiece of this research direction is a conjecture of Martin which asserts in a very strong way the unique nature of the Turing jump. Although Martin’s conjecture remains open, substantial partial results have been obtained which we view as strong evidence for its truth. While Martin’s conjecture paints a compelling picture, we will also discuss a conflicting possibility based on the difficulty of the problem of classifying the Turing degrees by invariants, as measured by the location of Turing equivalence in the hierarchy of countable Borel equivalence relations. A leitmotif of recursion theory is that degree structures such as the Turing degrees ought to be as rich and complex as possible. In the setting of Borel equivalence relations, the natural manifestation of this theme would be that Turing equivalence is a universal countable Borel equivalence relation. This would strongly contradict Martin’s conjecture.