## 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

David R. Belanger and Richard A. Shore

#### Abstract

We consider the set of *jumps below* a Turing degree, given by $\mathsf{JB}\left(\mathbf{a}\right)=\{\mathbf{x}\text{'}:\mathbf{x}\le \mathbf{a}\}$, with a focus on the problem: Which recursively enumerable (r.e.) degrees $\mathbf{a}$ are uniquely determined by $\mathsf{JB}\left(\mathbf{a}\right)$? 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 $\mathsf{JB}\left(\mathbf{a}\right)$, 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 $\mathsf{JB}\left({\mathbf{a}}_{0}\right)=\mathsf{JB}\left({\mathbf{a}}_{1}\right)$ within any possible jump class $\{\mathbf{x}:\mathbf{x}\text{'}=\mathbf{c}\}$. 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**

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