We consider the set of jumps below a Turing degree, given by , with a focus on the problem: Which recursively enumerable (r.e.) degrees are uniquely determined by ? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order of r.e. degrees. Namely, we show that if every high r.e. degree is determined by , then cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing pairs , of distinct r.e. degrees such that within any possible jump class . We give some extensions of the construction and suggest ways to salvage the attack on rigidity.
"On the Jumps of the Degrees Below a Recursively Enumerable Degree." Notre Dame J. Formal Logic 59 (1) 91 - 107, 2018. https://doi.org/10.1215/00294527-2017-0014