Bi-Isolated d.c.e. Degrees and Σ₁ Induction

2025/12/03 by Liu, Yong, Peng, Cheng
#03D28 #03F30 #03H15 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2512.03778

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 d is isolated if there exists a c.e. degree a<d such that every c.e. degree below d is also below a; d is upper isolated if there exists a c.e. degree a>d such that every c.e. degree above d is also above a; 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 IΣ₁.

Related

• Isolated d.c.e. degrees and Σ₁ induction
• Realizing Computably Enumerable Degrees in Separating Classes
• Finite final segments of the d.c.e. Turing degrees
• DNR and incomparable Turing degrees
• The isometry degree of a computable copy of ℓ^p