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The Banach-Tarski paradox Stan Wagon
Publisher: Cambridge University Press

I first encountered the Banach-Tarski paradox in one of these classes about 20 years ago, and it has been one of my white whales ever since. The Banach-Tarski Paradox says that you can take a sphere, cut it into a finite number of pieces, and then move and rotate these pieces around to make two new spheres just like the original. Banach and Tarski proved that, if the Axiom of Choice is assumed, a spherical ball can be cut into a finite number of pieces, and then reassembled into a ball of larger size. Technically, this isn't quite true in general, as there exist non-measurable subsets of the plane. After all, if one deals solely with finite sets, then there is no need to distinguish between countable and uncountable infinities, and Banach-Tarski type paradoxes cannot occur. In General Math is being discussed at Physics Forums. One of the interesting features of this branch of mathematics is the 'Axiom of choice' which to the beginner can seem to be the intuitive way to go, until they come to the 'Banach-Tarski paradox'. Question about the Banach–Tarski paradox. Using it, Bender got to make two, slightly smaller, copies of himself. The Banach-Tarski Dupla-Shrinker recently made an appearance on an episode of Futurama. This is essentially a continuous version of the Banach-Tarski paradox, and is therefore somewhat more horrifying than the standard version. In these notes I want to present a proof of the Banach-Tarski paradox, a consequence of the axiom of choice that shows us that a naive understanding of the concept of volume can lead to contradictions.