Reverse-engineering Reverse Mathematics

Abstract

An important open problem in Reverse Mathematics (Montalbán, 2011 [16]; Simpson, 2009 [25]) is the reduction of the first-order strength of the base theory from IΣ1 to IΔ0+exp. The system ERNA, a version of Nonstandard Analysis based on the system IΔ0+exp, provides a partial solution to this problem. Indeed, weak Königʼs lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA, while preserving the equivalences, but at the price of replacing equality with ‘≈’, i.e. infinitesimal proximity (Sanders, 2011 [19]). The logical principle corresponding to weak Königʼs lemma is the universal transfer principle from Nonstandard Analysis. Here, we consider the intermediate and mean value theorem and their uniform generalizations. We show that ERNAʼs Reverse Mathematics mirrors the situation in classical Reverse Mathematics. This further supports our claim from Sanders (2011) [19] that the Reverse Mathematics of ERNA plus universal transfer is a copy up to infinitesimals of that of WKL0. We discuss some of the philosophical implications of our results.