Covering the recursive sets

TL;DR

This paper addresses open questions in recursive set theory by providing new constructions and introducing the concept of i.o. subuniformity, revealing limitations of certain covering methods and exploring the measure-theoretic properties of classes.

Contribution

It introduces the concept of i.o. subuniformity, relates it to recursive measure theory, and constructs classes illustrating the limitations of non-probabilistic covering methods.

Findings

Existence of classes with recursive measure zero not covered by i.o. subuniformity.

Hyperimmune degrees can cover recursive sets, but some hyperimmune-free degrees cannot.

Probabilistic methods are necessary for certain classes that cannot be covered otherwise.

Abstract

We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales. At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover.