Axiom of choice: Difference between revisions

Lastly, and most famously, the axiom of choice implies the [[Banach-Tarski paradox]]. In particular, using the axiom of choice, it's possible to decompose any ball in 3D space into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. This is counterintuitive, but not truly paradoxical as the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points.