The Banach-Tarski Paradox

Abstract

In 1924, S. Banach and A. Tarski proved an astonishing, yet rather counterintuitive paradox: given a solid ball in $\mathbb{R}^3$, it is possible to partition it into finitely many pieces and reassemble them to form two solid balls, each identical in size to the first. When this paradox is applied to 3-dimensional space it does go against our intuition, but very often our intuition is flawed. The aim of the paper is to provide a comprehensive proof of the Banach-Tarski paradox, expanding in between the lines of the original volume. We explore the notions of paradoxical and equidecomposable sets which are phrased in terms of group actions. Finally, provided we have the Axiom of Choice at our disposal, we can construct sets that are nonmeasurable (not Lebesgue measurable) and the proof of the Banach-Tarski Paradox follows naturally.