On the Jumps of the Degrees Below a Recursively Enumerable Degree

Abstract

We consider the set of jumps below a Turing degree, given by $\mathsf{JB}\left(\mathbf{a}\right)=\{\mathbf{x}\text{'}:\mathbf{x}\le \mathbf{a}\}$, with a focus on the problem: Which recursively enumerable (r.e.) degrees $\mathbf{a}$ are uniquely determined by $\mathsf{JB}\left(\mathbf{a}\right)$? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order $\mathcal{R}$ of r.e. degrees. Namely, we show that if every high${}_2$ r.e. degree $\mathbf{a}$ is determined by $\mathsf{JB}\left(\mathbf{a}\right)$, then $\mathcal{R}$ cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing pairs ${\mathbf{a}}_0$, ${\mathbf{a}}_1$ of distinct r.e. degrees such that $\mathsf{JB}\left({\mathbf{a}}_0\right)=\mathsf{JB}\left({\mathbf{a}}_1\right)$ within any possible jump class $\{\mathbf{x}:\mathbf{x}\text{'}=\mathbf{c}\}$. We give some extensions of the construction and suggest ways to salvage the attack on rigidity.