Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 59, Number 1 (2018), 91-107.
On the Jumps of the Degrees Below a Recursively Enumerable Degree
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.
Notre Dame J. Formal Logic, Volume 59, Number 1 (2018), 91-107.
Received: 9 April 2015
Accepted: 18 June 2015
First available in Project Euclid: 20 July 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03D25: Recursively (computably) enumerable sets and degrees
Secondary: 03D28: Other Turing degree structures
Belanger, David R.; Shore, Richard A. On the Jumps of the Degrees Below a Recursively Enumerable Degree. Notre Dame J. Formal Logic 59 (2018), no. 1, 91--107. doi:10.1215/00294527-2017-0014. https://projecteuclid.org/euclid.ndjfl/1500537625