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arXiv · 1805.11342

Splittings and disjunctions in Reverse Mathematics

Sam Sanders
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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.

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Sam Sanders. 2018-05-29. Splittings and disjunctions in Reverse Mathematics. https://doi.org/10.1215/00294527-2019-0032

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