The uncountability of the reals and the Axiom of Choice

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.