The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere S2 (a 2-dimensional sphere in R3). It states that if a certain countable subset is removed from S2, then the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent. In particular, it follows that on S2 there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of B ∪ C is simultaneously 1/3 and 2/3 of the non-zero measure of the whole sphere).