University of Oslo MAT 2000 Project

In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible to partition the ball into finitely many pieces, reassemble them using rotations only, producing two new balls of the exact size as the original ball. In its strong form, the paradox states that for any two bounded sets A,B ∈ R with non-empty interior, it is possible to partition A, move the pieces around, and end up with B. For a paradoxical decomposition of the sphere, it can be shown that 4 pieces actually is enough. This short paper aims to prove the BanachTarski paradox in its weak and strong forms, together with the justmentioned lower bound on the number of pieces in the decomposition.