The AC was, for a long time, a controversial axiom. On the one hand, it is very useful and of wide use in mathematics. On the other hand, it has rather unintuitive consequences, such as the Banach-Tarski Paradox, which says that the unit ball can be partitioned into finitely-many pieces, which can then be rearranged to form two unit balls. The objections to the axiom arise from the fact that it asserts the existence of sets that cannot be explicitly defined. But Gödel’s 1938 proof of its consistency, relative to the consistency of ZF, dispelled any suspicions left about it.