On the degrees of constructively immune sets

TL;DR

This paper explores the properties and applications of constructively immune sets, including their Turing degrees and relation to numberings of rationals, expanding understanding of their role in computability theory.

Contribution

It introduces new applications of constructively immune sets in numberings of rationals and analyzes their Turing degrees and connections to dense sets.

Findings

Constructively immune sets relate to numberings of rationals.

Analysis of Turing degrees of these sets.

Connections to $\, ext{Sigma}^0_1$-dense sets.

Abstract

Xiang Li (1983) introduced what are now called constructively immune sets as an effective version of immunity. Such have been studied in relation to randomness and minimal indices, and we add another application area: numberings of the rationals. We also investigate the Turing degrees of constructively immune sets and the closely related $\Sigma^0_1$-dense sets of Ferbus-Zanda and Grigorieff (2008).