On the Jumps of the Degrees Below a Recursively Enumerable Degree

Abstract

We consider the set of jumps below a Turing degree, given by JB(a)={x':x≤a}, with a focus on the problem: Which recursively enumerable (r.e.) degrees a are uniquely determined by JB(a)? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high2 r.e. degree a is determined by JB(a), then R cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing pairs a0, a1 of distinct r.e. degrees such that JB(a0)=JB(a1) within any possible jump class {x:x'=c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity.