  Although the usefulness of AC quickly become clear, doubts about its soundness remained. These doubts were reinforced by the fact that it had certain strikingly counterintuitive consequences.  The most spectacular of these was Banach and Tarski’s paradoxical decompositions of the sphere (Banach and Tarski 1924): any solid sphere can be split into finitely many pieces which can be reassembled to form two solid spheres of the same size; and any solid sphere can be split into finitely many pieces in such a way as to enable them to be reassembled to form a solid sphere of arbitrary size. (See Wagon 1993.) 