Independence and Induction in Reverse Mathematics

TL;DR

This paper explores the logical strength and relationships of principles related to the existence of large, mutually independent families of objects, such as almost disjoint sets and eventually different functions, within reverse mathematics.

Contribution

It characterizes the induction strength needed for these principles and establishes equivalences with other combinatorial principles in reverse mathematics.

Findings

Over certain induction levels, $ eg ext{MAD}$ is equivalent to the $ ext{DOM}$ principle.

The study reveals the importance of $ ext{B} ext{ extSigma}_2^0$ and $ ext{I} ext{ extSigma}_2^0$ levels in the strength of these principles.

Relations between $ ext{MAD}$, $ ext{MED}$, and their variants are clarified.

Abstract

We continue the project of the study of reverse mathematics principles inspired by cardinal invariants. In this article in particular we focus on principles encapsulating the existence of large families of objects that are in some sense mutually independent. More precisely, we study the principle $\mathsf{MAD}$ stating that a maximal family of pairwise almost disjoint sets exists; and the principle $\mathsf{MED}$ expressing the existence of a maximal family of functions that are pairwise eventually different. We investigate characterisations of and relations between these principles and some of their variants. It turns out that induction strength at the levels of $\mathsf{B}\mathrm{\Sigma}_2^0$ or $\mathsf{I}\mathrm{\Sigma}_2^0$ is an essential parameter; for instance, over $\mathsf{B}\mathrm{\Sigma}_2^0$, we show that $\neg\mathsf{MAD}$ is equivalent to the principle $\mathsf{DOM}$ expressing that every weakly represented family of functions is dominated by some other function.