On The Computability of Perfect Subsets of Sets with Positive Measure

TL;DR

This paper investigates the computability properties of perfect subsets within positive measure sets, revealing their potential for weak computational strength and exploring their implications in reverse mathematics.

Contribution

It demonstrates that perfect subsets of positive measure sets can have weak computational strength and links their existence to reverse mathematics.

Findings

Perfect subsets can have weak computational strength.

Existence of perfect subsets relates to reverse mathematics.

Provides insights into the computability of measure-theoretic sets.

Abstract

A set $X \subseteq 2^\omega$ with positive measure contains a perfect subset. We study such perfect subsets from the viewpoint of computability and prove that these sets can have weak computational strength. Then we connect the existence of perfect subsets of sets with positive measure with reverse mathematics.