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

TL;DR

This paper investigates the properties of isolated and upper isolated d.c.e. degrees within reverse recursion theory, establishing their connection to $oldsymbol{ ext{I}}oldsymbol{ ext{S}}oldsymbol{ ext{i}}oldsymbol{ ext{m}}oldsymbol{ ext{a}}oldsymbol{ ext{1}}$ induction through formal logical systems.

Contribution

It demonstrates the equivalence between the existence of isolated degrees and $oldsymbol{ ext{I}}oldsymbol{ ext{S}}oldsymbol{ ext{i}}oldsymbol{ ext{m}}oldsymbol{ ext{a}}oldsymbol{ ext{1}}$ induction in a formal logical framework.

Findings

Existence of an isolated proper d.c.e. degree below $oldsymbol{0'}$ is equivalent to $oldsymbol{ ext{I}}oldsymbol{ ext{S}}oldsymbol{ ext{a}}oldsymbol{ ext{1}}$ induction.

Existence of an upper isolated proper d.c.e. degree below $oldsymbol{0'}$ is also equivalent to $oldsymbol{ ext{I}}oldsymbol{ ext{S}}oldsymbol{ ext{a}}oldsymbol{ ext{1}}$ induction.

The results connect degree-theoretic properties with formal logical induction principles.

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 $\mathbf{0}'$.