Splittings and Disjunctions in Reverse Mathematics
Sam Sanders · Notre Dame Journal of Formal Logic · 2019
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↔(B∧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↔(E∨F), 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.