Banach–Tarski paradox

The Banach–Tarski paradox is a theorem in set-theoretic geometry that states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets that can 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 original shape. The pieces themselves are not "solids" in the traditional sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.

An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".

The theorem is a veridical paradox: it contradicts basic geometric intuition, but is not false or self-contradictory. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volume is not mathematically absurd and is even included in the formal definition of volume. But this is not applicable here because in this case it is impossible to define the volumes of the considered subsets.