## Notre Dame Journal of Formal Logic

### On the Jumps of the Degrees Below a Recursively Enumerable Degree

#### Abstract

We consider the set of jumps below a Turing degree, given by $\mathsf{JB}(\mathbf{a})=\{\mathbf{x}':\mathbf{x}\leq\mathbf{a}\}$, with a focus on the problem: Which recursively enumerable (r.e.) degrees $\mathbf{a}$ are uniquely determined by $\mathsf{JB}(\mathbf{a})$? 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 $\mathsf{JB}(\mathbf{a})$, 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 $\mathsf{JB}(\mathbf{a}_{0})=\mathsf{JB}(\mathbf{a}_{1})$ within any possible jump class $\{\mathbf{x}:\mathbf{x}'=\mathbf{c}\}$. 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
Accepted: 18 June 2015
First available in Project Euclid: 20 July 2017

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

#### 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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