A Brief History
The Banach-Tarski Paradox was first published in 1924 by Banach and Tarski in their groundbreaking paper, “Sur la décomposition des ensembles de points en parties respectivement congruentes” (On the decomposition of sets of points into congruent parts). Their work was based on earlier discoveries by Austrian mathematician Felix Hausdorff, who had demonstrated a similar paradoxical decomposition for certain subsets of Euclidean space. Banach and Tarski extended Hausdorff’s work to prove their astonishing result for the 3-dimensional sphere.
The Paradox
The Banach-Tarski Paradox is a consequence of the Axiom of Choice, an important and controversial principle in set theory. While the Axiom of Choice has many useful applications in mathematics, it also leads to some perplexing results, such as the Banach-Tarski Paradox.