Hausdorff paradox
The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere . It states that if a certain countable subset is removed from, then the remainder can be divided into three disjoint subsets and such that and are all congruent. In particular, it follows that on there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal.
The paradox was published in Mathematische Annalen in 1914 and also in Hausdorff's book, Grundzüge der Mengenlehre, the same year. The proof of the much more famous Banach–Tarski paradox uses Hausdorff's ideas. The proof of this paradox relies on the Axiom of Choice.
This paradox shows that there is no finitely additive measure on a sphere defined on all subsets which is equal on congruent pieces. The structure of the group of rotations on the sphere plays a crucial role here the statement is not true on the plane or the line. In fact, as was later shown by Banach, it is possible to define an "area" for all bounded subsets in the Euclidean plane in such a way that congruent sets will have equal "area". This implies that if two open subsets of the plane are equi-decomposable then they have equal area.
