Martin's conjecture, arithmetic equivalence, and 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 We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation.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.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 DOI:
 10.48550/arXiv.1109.1875
 arXiv:
 arXiv:1109.1875
 Bibcode:
 2011arXiv1109.1875M
 Keywords:

 Mathematics  Logic
 EPrint:
 Corrected typos