The Banach-Tarski Paradox

TL;DR

The paper provides a comprehensive proof of the Banach-Tarski paradox, demonstrating how a solid ball can be partitioned and reassembled into two identical balls using group actions and the Axiom of Choice.

Contribution

It offers an expanded, detailed proof of the Banach-Tarski paradox, clarifying the concepts of paradoxical and equidecomposable sets within the framework of group actions.

Findings

The paradoxical decomposition relies on the Axiom of Choice.

Sets involved are nonmeasurable and not Lebesgue measurable.

The proof illustrates the counterintuitive nature of volume in set theory.

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.