Splittings and disjunctions in Reverse Mathematics

TL;DR

This paper investigates the phenomena of splittings and disjunctions in Reverse Mathematics, revealing many natural examples in higher-order RM and discussing their significance in the foundations of mathematics.

Contribution

It provides the first extensive analysis of splittings and disjunctions in higher-order RM, contrasting with their rarity in second-order RM.

Findings

Many natural splittings and disjunctions are found in higher-order RM.

Examples of theorems that can be expressed as conjunctions or disjunctions of independent parts.

Discussion on the implications of these phenomena for the foundations of mathematics.

Abstract

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, i.e. non-set-theoretic, mathematics. As suggested by the title, this paper deals with two (relatively rare) RM-phenomena, namely splittings and disjunctions. As to splittings, there are some examples in RM of theorems $A, B, C$ such that $A\leftrightarrow (B\wedge C)$, i.e. $A$ can be split into two independent (fairly natural) parts $B$ and $C$. As to disjunctions, there are (very few) examples in RM of theorems $D, E, F$ such that $D\leftrightarrow (E\vee F)$, i.e. $D$ can be written as the disjunction of two independent (fairly natural) parts $E$ and $F$. By contrast, we show in this paper that there is a plethora of (natural) splittings and disjunctions in Kohlenbach's higher-order RM. Finally, we discuss the role of these results in the grand scheme of things.