Banach-Tarski Balls

First stated by Stefan Banach and Alfred Tarski in 1924, the Banach-Tarski paradox is the famous "doubling the ball" paradox, which states that by using the axiom of choice it is possible to take a solid ball in 3-dimensional space, cut it up into finitely many (non-measurable) pieces, and moving them using only rotations and translations, reassemble the pieces into two balls of the same radius as the original. Banach and Tarski intended for this proof to demonstrate that the axiom of choice was incorrect, but the nature of the proof is such that most mathematicians take it to mean that the axiom of choice merely results in bizarre and unintuitive consequences. In other words, a marble could be cut up into finitely many pieces and reassembled into a planet, or a telephone could be transformed into a water lily. These transformations are not possible with real objects made of a finite number of atoms and bounded volumes, but it is possible with their geometric shapes. What makes the paradox possible is that the pieces are infinitely convoluted. Technically, they are not measurable, and so they do not have "reasonable" boundaries or a "volume" in the ordinary sense. It is impossible to carry out such a disassembly physically because disassembly "with a knife" can create only measurable sets. This pure existence statement in mathematics points out that there are many more sets than just the measurable sets familiar to most people.