Bi-Isolated d.c.e. Degrees and $\Sigma_1$ Induction

TL;DR

This paper proves the existence of bi-isolated d.c.e. degrees within models of , advancing understanding of the structure of Turing degrees and their properties in computability theory.

Contribution

It establishes the existence of bi-isolated d.c.e. degrees in models of , a novel result in the study of Turing degrees and their isolation properties.

Findings

Existence of bi-isolated d.c.e. degrees proven

Bi-isolated degrees exist in models of 

Advances understanding of Turing degree structure

Abstract

A Turing degree is d.c.e. if it contains a set that is the difference of two c.e. sets. A d.c.e. degree $\mathbf{d}$ is isolated if there exists a c.e. degree $\mathbf{a}<\mathbf{d}$ such that every c.e. degree below $\mathbf{d}$ is also below $\mathbf{a}$; $\mathbf{d}$ is upper isolated if there exists a c.e. degree $\mathbf{a}>\mathbf{d}$ such that every c.e. degree above $\mathbf{d}$ is also above $\mathbf{a}$; $\mathbf{d}$ is bi-isolated if it is both isolated and upper isolated. In this paper, we prove the existence of bi-isolated d.c.e. degrees in models of $\mathsf{I}\Sigma_1$.