Uniform and nonstandard existence in Reverse Mathematics

TL;DR

This paper proposes an alternative classification of theorems in Reverse Mathematics based on Feferman's Explicit Mathematics, connecting standard and nonstandard existence through computability and functional interpretation.

Contribution

It introduces the Explicit Mathematics theme (EMT) in Reverse Mathematics, linking theorem existence to computability via nonstandard analysis and extending the Big Five systems.

Findings

Established EMT examples across the Big Five systems.

Demonstrated the equivalence between standard existence and computability.

Extended Reverse Mathematics with higher types and nonstandard analysis.

Abstract

Reverse Mathematics is a program in the foundations of mathematics which provides an elegant classification of theorems of ordinary mathematics based on computability. Our aim is to provide an alternative classification of theorems based on the central tenet of Feferman's Explicit Mathematics, namely that a proof of existence of an object yields a procedure to compute said object. Our classification gives rise to the Explicit Mathematics theme (EMT) of Nonstandard Analysis. Intuitively speaking, the EMT states that a standard object with certain properties can be computed by a functional if and only if this object exists classically with these same standard and nonstandard properties. In this paper, we establish examples for the EMT ranging from the weakest to the strongest Big Five system of Reverse Mathematics. Our results are proved over the usual base theory of Reverse Mathematics, conservatively extended with higher types and Nelson's internal approach to Nonstandard Analysis.