Notre Dame Journal of Formal Logic

On the Jumps of the Degrees Below a Recursively Enumerable Degree

David R. Belanger and Richard A. Shore

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Abstract

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 1 (2018), 91-107.

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

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1500537625

Digital Object Identifier
doi:10.1215/00294527-2017-0014

Mathematical Reviews number (MathSciNet)
MR3744353

Zentralblatt MATH identifier
06848193

Subjects
Primary: 03D25: Recursively (computably) enumerable sets and degrees
Secondary: 03D28: Other Turing degree structures

Keywords
r.e. degrees Turing jump rigidity problem

Citation

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


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References

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