Why is Banach-Tarski Paradox a paradox? I mean, it just proves that the set of points on a sphere has the same carnality as the set of points of two spheres. This isn't something contra-intuitive once the open set (0, 1) has the same carnality as the set of real numbers. Does mathematicians call anything that contradicts an average person intuition a paradox? Paradox isn't a synonym for contradiction?