Szego Seminar: "Duplicating a Ball: A Proof for Banach-Tarski Paradox"

Abstract: We partition (or better to say prove that it can be partitioned) the unit open ball of the 3-dimensional Euclidean space into finitely many subsets, which can then be reassembled back, using only finitely many rotations and translations, to yield two identical copies of the original ball! (Of course some of these pieces are non-Lebesgue measurable.) This is called the (Hausdorff-)Banach-Tarski paradox, and its proof uses the Axiom of Choice in an essential way (essential in the sense that the paradox disappears in Zermelo-Frankel set theory). We give a complete proof due to Waclaw Sierpinski. Two essential ideas are: (1) the group of isometries of the Euclidean space contains a free group of two generators; (2) Hilbert's infinite hotel trick.