Reverse Mathematics of Brouwer's continuity theorem and related principles

TL;DR

This paper analyzes Brouwer's Continuity Theorem and related principles using Reverse Mathematics and Nonstandard Analysis, proposing an alternative classification based on explicit computability of objects.

Contribution

It introduces an Explicit Mathematics-based classification (EMT) for intuitionistic principles, linking existence proofs to computability within a higher-type and nonstandard framework.

Findings

Established EMT for several intuitionistic principles

Provided a new classification of theorems based on explicit computability

Connected classical and nonstandard properties through EMT

Abstract

In intuitionistic mathematics, the Brouwer Continuity Theorem states that all total real functions are (uniformly) continuous on the unit interval. We study this theorem and related principles from the point of view of Reverse Mathematics over a base theory accommodating higher types and Nonstandard Analysis. With regard to the bigger picture, Reverse Mathematics provides a 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). 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. Hence, we establish the EMT for a series of intuitionistic principles in this paper.