The Banach-Tarski Paradox

Set Theory · Axiom Academy

LESSON The Banach-Tarski Paradox One of the most counterintuitive consequences of the Axiom of Choice The Banach-Tarski theorem proves something extraordinary: a ball of radius 1 can be decomposed into finitely many pieces (typically 5) and reassembled by rotations and translations into two balls, each of radius 1. This seems to violate basic physical intuitions about volume conservation. How can one ball become two balls of the same size? 2. Role of the Axiom of Choice The key to the paradox lies in the nature of the pieces. The decomposition uses the Axiom of Choice to construct sets that are not Lebesgue measurable. The construction uses the free group on two generators acting on the sphere, creating orbits whose representatives are chosen using AC. These pieces are so pathological that they cannot have a well-defined volume. 3. Why This Doesn't Violate Physics The Banach-Tarski paradox does not contradict physical laws for several crucial reasons: The pieces are not measurable: They have no volume, so conservation of volume is not violated Infinitely complex: The pieces require choosing from infinitely many points with no constructive pattern Not physically realizable: Matter is made of discrete atoms, not mathematical points Quantum mechanics: The uncertainty principle prevents such precise decompositions The paradox operates in the realm of pure mathematics, where sets of points can be manipulated in ways impossible for physical objects. 4. The Pieces Are Not Measurable

This is the written version of the interactive lesson above. See the full Set Theory course.