Isolated d.c.e. degrees and $\Sigma_1$ induction

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 by a c.e. degree $\mathbf{a}<\mathbf{d}$ if all c.e. degrees that are below $\mathbf{d}$ are also below $\mathbf{a}$; $\mathbf{d}$ is isolated from above by a c.e. degree $\mathbf{a}>\mathbf{d}$ if all c.e. degrees that are above $\mathbf{d}$ are also above $\mathbf{a}$. In this paper, we study the inductive strength of both isolated and upper isolated d.c.e. degrees from the point of view of reverse recursion theory. We show that (1) $P^{-} + B\Sigma_1 + \text{Exp} \vdash I\Sigma_1 \leftrightarrow$ There is an isolated proper d.c.e. degree below $\mathbf{0}'$; (2) $P^{-} + B\Sigma_1 + \text{Exp} \vdash I\Sigma_1 \leftrightarrow$ There is an upper isolated proper d.c.e. degree below $\mathbf{0}'$.