Reverse Mathematics of the uncountability of $\mathbb{R}$: Baire classes, metric spaces, and unordered sums

TL;DR

This paper explores the logical and computational aspects of the uncountability of real numbers, connecting principles like NIN and NBI with theorems on Baire classes, metric spaces, and unordered sums within higher-order Reverse Mathematics.

Contribution

It extends the study of NIN and NBI principles by linking them to classical theorems in analysis, revealing their strength and relationships within a continuity-based logical framework.

Findings

Unordered sums are equivalent to countable series when they exist.

NIN with the Cauchy criterion implies the uncountability of reals.

NBI with limits implies the uncountability of reals.

Abstract

Dag Normann and the author have recently initiated the study of the logical and computational properties of the uncountability of $\mathbb{R}$ formalised as the statement $\textsf{NIN}$ (resp. $\textsf{NBI}$ that there is no injection (resp. bijection) from $[0,1]$ to $\mathbb{N}$. On one hand, these principles are hard to prove relative to the usual scale based on comprehension and discontinuous functionals. On the other hand, these principles are among the weakest principles on a new complimentary scale based on (classically valid) continuity axioms from Brouwer's intuitionistic mathematics. We continue the study of $\textsf{NIN}$ and $\textsf{NBI}$ relative to the latter scale, connecting these principles with theorems about Baire classes, metric spaces, and unordered sums. The importance of the first two topics requires no explanation, while the final topic's main theorem, i.e. that when they exist, unordered sums are (countable) series, has the rather unique property of implying $\textsf{NIN}$ formulated with the Cauchy criterion, and (only) $\textsf{NBI}$ when formulated with limits. This study is undertaken within Ulrich Kohlenbach's framework of higher-order Reverse Mathematics.