Refining the taming of the Reverse Mathematics zoo

TL;DR

This paper refines the understanding of the Reverse Mathematics zoo by showing that many exceptional theorems, when uniformized, fall into the Big Five categories and can be derived using intuitionistic logic, connecting nonstandard and explicit equivalences.

Contribution

It extends previous results to recent zoo theorems, demonstrating their uniform versions fall into the arithmetical comprehension category and can be proved with intuitionistic logic, linking nonstandard and explicit forms.

Findings

Uniform versions of zoo-theorems fall into the arithmetical comprehension category.

Explicit equivalences can be derived using only intuitionistic logic.

Nonstandard equivalences can be recovered from explicit ones.

Abstract

Reverse Mathematics is a program in the foundations of mathematics. It provides an elegant classification in which the majority of theorems of ordinary mathematics fall into only five categories, based on the 'Big Five' logical systems. Recently, a lot of effort has been directed towards finding exceptional theorems, i.e. which fall outside the Big Five. The so-called Reverse Mathematics zoo is a collection of such exceptional theorems (and their relations). It was shown in [17] that a number of uniform versions of the zoo-theorems, i.e. where a functional computes the objects stated to exist, fall in the third Big Five category arithmetical comprehension, inside Kohlenbach's higher-order Reverse Mathematics. In this paper, we extend and refine the results from [17]. In particular, we establish analogous results for recent additions to the Reverse Mathematics zoo, thus establishing that the latter disappear at the uniform level. Furthermore, we show that the aforementioned equivalences can be proved using only intuitionistic logic. Perhaps most surprisingly, these explicit equivalences are extracted from nonstandard equivalences in Nelson's internal set theory, and we show that the nonstandard equivalence can be recovered from the explicit ones.