Pathwise-random trees and models of second-order arithmetic

TL;DR

This paper investigates the properties of pathwise-random trees, showing limitations on their computational power and constructing models of second-order arithmetic that distinguish certain compactness principles.

Contribution

It introduces new results on the computational limitations of pathwise-random trees and constructs models of second-order arithmetic separating specific principles.

Findings

No weakly 2-random real computes a perfect pathwise-random tree.

Existence of a positive-measure pathwise-random tree that does not compute any PA extension.

Existence of a perfect pathwise-random tree that does not compute any tree of positive measure with finite randomness deficiency.

Abstract

A tree is pathwise-random if all of its paths are Martin-Lof random. We show that (a) no weakly 2-random real computes a perfect pathwise-random tree; it follows that the class of perfect pathwise-random trees is null, with respect to any computable measure; (b) there exists a positive-measure pathwise-random tree which does not compute any complete extension of Peano arithmetic; and (c) there exists a perfect pathwise-random tree which does not compute any tree of positive measure and finite randomness deficiency. We then obtain models of second-order arithmetic that separate compactness principles below weak Konigs lemma, answering questions by Chong et al.(2019).