The Banach-Tarski Paradox is probably the most counter-intuitive theorem in any field of mathematics. It states, in layman's terms, that a solid ball in R_ can be split into finitely non-overlapping pieces, which can then be put back together in a different way only using rigid motions to produce two identical copies of the original ball. A stronger version of this paradox is often stated as "a pea can be chopped up and reassembled into the sun." This is possible to prove because of Banach and Tarski's use of the Axiom of Choice. Because of the increase in measure, these paradoxical decompositions must involve nonmeasurable sets. Measurable sets and sets with the property of Baire have a lot of similar characteristics, so it is natural to ask if the Banach-Tarski Paradox is possible using pieces that have the property of Baire. Surprisingly, as we will see, this is also possible.
