Splittings and Disjunctions in Reverse 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, that is, 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 $A\leftrightarrow (B\wedge C)$, that is, $A$ 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 $D\leftrightarrow \left(E\vee F\right)$, that is, $D$ 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.