Bi-Isolated d.c.e. Degrees and $Σ_1$ Induction
Yong Liu, Cheng Peng
TL;DR
The paper proves the existence of bi-isolated d.c.e. degrees within models of $I\Sigma_1$, resolving a question about the strength of induction needed for such degrees. It constructs a d.c.e. set $D$ and a c.e. set $A\le_T D$ with $A<_T D<_T K$, where $K=\{2e+1: \Phi_e(e)\downarrow\}$, using a finite-injury priority argument organized on a priority tree with $N_e$, $R_e$, and $P_e$ strategies. Key innovations combine Friedberg–Muchnik-style isolation ideas with Lachlan-set machinery and Cycle modules to ensure $D$ is isolated from below by $A$ and from above by $K$, while maintaining $D=L(D)$ and controlling injuries to keep the construction within $I\Sigma_1$ bounds. The work also discusses how these results relate to reverse mathematics, the Sacks Density Theorem, and related open problems in weaker induction frameworks.
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 $\mathsf{I}Σ_1$.