4 research outputs found
Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations
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 We will present a long unpublished result of Slaman and Steel that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.Comment: Corrected typo
Part 1 of Martin's Conjecture for order-preserving and measure-preserving functions
Martin's Conjecture is a proposed classification of the definable functions
on the Turing degrees. It is usually divided into two parts, the first
classifies functions which are not above the identity and the second of
classifies functions which are above the identity. Slaman and Steel proved the
second part of the conjecture for Borel functions which are order-preserving
(i.e. which preserve Turing reducibility). We prove the first part of the
conjecture for all order-preserving functions. We do this by introducing a
class of functions on the Turing degrees which we call "measure-preserving" and
proving that part 1 of Martin's Conjecture holds for all measure-preserving
functions and also that all non-trivial order-preserving functions are
measure-preserving. Our result on measure-preserving functions has several
other consequences for Martin's Conjecture, including an equivalence between
part 1 of the conjecture and a statement about the structure of the
Rudin-Keisler order on ultrafilters on the Turing degrees.Comment: 44 pages; updated to correct some attributions and fix some typo