Open Access
2018 On the Jumps of the Degrees Below a Recursively Enumerable Degree
David R. Belanger, Richard A. Shore
Notre Dame J. Formal Logic 59(1): 91-107 (2018). DOI: 10.1215/00294527-2017-0014


We consider the set of jumps below a Turing degree, given by JB(a)={x':xa}, 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.


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David R. Belanger. Richard A. Shore. "On the Jumps of the Degrees Below a Recursively Enumerable Degree." Notre Dame J. Formal Logic 59 (1) 91 - 107, 2018.


Received: 9 April 2015; Accepted: 18 June 2015; Published: 2018
First available in Project Euclid: 20 July 2017

zbMATH: 06848193
MathSciNet: MR3744353
Digital Object Identifier: 10.1215/00294527-2017-0014

Primary: 03D25
Secondary: 03D28

Keywords: r.e. degrees , rigidity problem , Turing jump

Rights: Copyright © 2018 University of Notre Dame

Vol.59 • No. 1 • 2018
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