Banach–Tarski paradox

The Banach–Tarski paradox, a mind-bending concept in set theory, asserts that a solid sphere can be divided into a finite number of peculiar subsets and reassembled to form two identical copies of the original. This process, involving only rotations and translations, seemingly defies intuition and conservation of matter. The paradox hinges on the axiom of choice and the existence of non-measurable sets, utilizing pieces that are dense, yet scattered point sets rather than conventional solids. Remarkably, this counterintuitive duplication can be achieved using just five such subsets.