A Gentle Introduction to the Axiom of Choice

TL;DR

This paper surveys the axiom of choice ($\mathsf{AC}$), tracing its origins from Cantor and Zermelo and presenting its formal statement as the existence of a choice function for any family of nonempty sets. It discusses two main objections, nonconstructivity and counterintuitive consequences, and then articulates three rationale types for accepting $\mathsf{AC}$: its central role in many areas of mathematics, its ability to simplify proofs, and the relative consistency results that position $\mathsf{AC}$ within the foundational framework of set theory. The discussion covers key consequences such as the Banach–Tarski paradox, nonmeasurable sets, and the equivalences to the well ordering principle and Zorn's lemma, while also outlining the Gödel–Cohen program for relative consistency. The overall aim is to provide non experts with a balanced perspective on the necessity, impact, and philosophical status of $\mathsf{AC}$ in modern mathematics.

Abstract

This article offers a gentle introduction to the axiom of choice. We introduce the axiom, discuss some common objections to it, and present three kinds of reasons to accept it. Although the exposition is aimed at non-experts in set theory, we also include some lesser-known results.