# Banach-Tarski Paradox

**Type:** Theorem  
**Domain:** Named Results / Foundations & Logic  
**Proved:** 1924, Stefan Banach and Alfred Tarski  
**Codex URL:** /mathematics/named-results/banach-tarski-paradox/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A solid ball can be decomposed into finitely many pieces and reassembled into two balls identical to the original — a genuine theorem, not a contradiction.

## Why it's not a real paradox
No logical contradiction — a fully valid theorem from accepted axioms, just wildly counterintuitive.

## Non-measurable sets
The "pieces" have no well-defined volume — infinitely intricate, impossible with real physical matter.

## Role of the Axiom of Choice
Requires infinitely many arbitrary selections with no explicit rule — exactly what AC guarantees. Without AC (in some models), every set IS measurable and the paradox vanishes.

## Minimum pieces: exactly 5
Proven minimum; 4 pieces are never enough.

## Hausdorff's 1914 precursor
A decade earlier, involving a sphere with countably many points removed — Banach-Tarski extended this to the full solid ball.

## Group-theoretic mechanism
Exploits a free subgroup within the 3D rotation group — the essential engine behind the construction.

## Solovay's 1970 model
Assuming a large cardinal, constructs a consistent model where every set of reals is measurable — confirming AC is genuinely necessary for the paradox.

## Sources
### Tier 1
- Banach, S. and Tarski, A. (1924). Fundamenta Mathematicae.
- Wagon, S. (1985). The Banach-Tarski Paradox.
### Tier 2
- Solovay, R.M. (1970). Annals of Mathematics.

---
*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*