The Banach-Tarski paradox

Abstract

In this thesis we present a proof of the Banach-Tarski paradox, a counterintuitive result that states that any ball in R3 can be cut into finitely many pieces and then be reassembled into two copies of the original ball. Since the result follows from the axiom of choice it is important for assessing its role as an axiom of mathematics. A related result that we also include is that the minimal number of pieces in such a decomposition of any ball in R3 is five. The proof uses the paradoxicality of the free group on two generators and the existence of a free subgroup of the special orthogonal group SO3. We also give a proof of Tarski’s theorem, which states that the existence of a finitely additive, isometry invariant measure normalizing a set is equivalent to that set not being paradoxical. The proof makes use of the Hahn-Banach theorem and relies on the concept of a group acting on several copies of a set.