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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.

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Autor(es)

Belanger, David R. -  Shore, Richard A. - 

Id.: 70828094

Idioma: inglés  - 

Versión: 1.0

Estado: Final

Tipo:  application/pdf - 

Palabras claver.e. degrees - 

Tipo de recurso: Text  - 

Tipo de Interactividad: Expositivo

Nivel de Interactividad: muy bajo

Audiencia: Estudiante  -  Profesor  -  Autor  - 

Estructura: Atomic

Coste: no

Copyright: sí

: Copyright 2018 University of Notre Dame

Formatos:  application/pdf - 

Requerimientos técnicos:  Browser: Any - 

Relación: [References] 0029-4527
[References] 1939-0726

Fecha de contribución: 06-ene-2018

Contacto:

Localización:
* Notre Dame J. Formal Logic 59, no. 1 (2018), 91-107
* doi:10.1215/00294527-2017-0014

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