The Banach–Tarski Paradox
The Banach–Tarski Paradox
Thesis
Banach–Tarski: a ball in 3D can be split into finitely many pieces and rigidly reassembled into two balls of the same size. Pieces are non-measurable; the proof uses the Axiom of Choice. Conceptual chapter — not a physical claim about oranges.
Mental model
ball → AC pieces → isometries → two balls
volume only for measurable setsWorked examples
What fails without Choice / in 2D
Analogous finite decompositions fail in \(\mathbb{R}^2\) with isometries; without Choice the pieces need not exist.
Where Choice shows up
Free groups inside \(SO(3)\) and Choice-picked orbits yield non-measurable pieces (see Wagon).
Trap
Using Banach–Tarski as an engineering gotcha. It does not break energy conservation; volume is undefined for those pieces.
Boring rule
Stay in Lebesgue-measurable sets when you need volume.
Try this
- Read the free-group paradox (Hausdorff).
- Compare Vitali sets.
- Skim Wagon, The Banach–Tarski Paradox.
Sources
- Wagon; measure-theory texts