The uncountability of the reals and the Axiom of Choice

TL;DR

This paper explores the logical strength of the uncountability of the reals, showing that it cannot be proven in certain strong systems and relating it to fragments of the Axiom of Choice.

Contribution

It demonstrates that the uncountability principle $ extsf{NIN}_{[0,1]}$ is independent of strong logical systems and connects it to fragments of the Axiom of Choice within higher-order Reverse Mathematics.

Findings

Strong logical systems cannot prove $ extsf{NIN}_{[0,1]}$.

Such systems imply second-order arithmetic and fragments of the Axiom of Choice.

The study relates uncountability to fragments of the Axiom of Choice in Kohlenbach's framework.

Abstract

The uncountability of the reals was first established by Cantor in what was later heralded as the first paper on set theory. Since the latter constitutes the official foundations of mathematics, the logical study of the uncountability of the reals is a worthy endeavour for historical, foundational, and conceptual reasons. In this paper, we shall study the following principle: $\textsf{NIN}_{[0,1]}$: there is no injection from the unit interval to the natural numbers. We show that relatively strong logical systems cannot prove $\textsf{NIN}_{[0,1]}$. In particular, the former system implies second-order arithmetic and fragments of the Axiom of Choice, including dependent choice. We also study the latter choice fragments in Kohlenbach's higher-order Reverse Mathematics.