The Biggest Five of Reverse Mathematics

TL;DR

This paper extends the Big Five phenomenon in Reverse Mathematics by unifying second-order and third-order approaches, establishing numerous equivalences, and exploring the boundaries of the Big Five systems with various theorems.

Contribution

It unifies two approaches to Reverse Mathematics and broadens the scope of the Big Five phenomenon through new equivalences and analysis of third-order theorems.

Findings

Many second-order Big Five systems are equivalent to third-order theorems in analysis.

Certain generalizations of third-order theorems fall outside the Big Five.

The unification reveals deeper connections between different levels of mathematical logic.

Abstract

The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five systems together are called the 'Big Five'. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalisations and variations of the aforementioned third-order theorems fall far outside of the Big Five.