The K-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [MN06, Question 4.2] and later in [Nie09, Problem 5.5.8]. We give a negative answer to this question. In fact, we show the followi...

This paper will study sets and degrees containing sets that are determined as limits of computable approximations. By the Shoenfield Limit Lemma, the limit computable sets are precisely the degrees below ∅′. In particular, the paper will study limit computable sets by classifying them according to bounds to the number of changes to elements in various approximations of the sets. This leads to t...

Assuming ZFC, we prove that CH holds if and only if there exists a cofinal maximal chain of order type ω1 in the Turing degrees.

In this article we give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture on Borel functions from an analytic subset of a Polish space into a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to prove that several prominent results in recursion t...

by Stephen M. Walk We focus on a particular class of computably enumerable (c.e.) degrees, the array noncomputable degrees defined by Downey, Jockusch, and Stob, to answer questions related to array noncomputable degrees and definability in the partial ordering (R,≤) of c.e. degrees. First we demonstrate that the lattice M5 cannot be embedded into the c.e. degrees below every array noncomputabl...