Abstract
In ZFC, a solid ball in ℝ3 can be partitioned into finitely many non-measurable pieces and reassembled into two balls congruent to the original. This page summarizes the statement, the role of the Axiom of Choice, amenability (2D vs 3D), and routes to avoid BT by changing axioms or restricting sets.
Contents
1. Theorem (ZFC)
Banach–Tarski. In ℝ3, a ball can be partitioned into finitely many pairwise disjoint sets and reassembled via rotations/translations into two balls congruent to the original.
Figure — Schematic only: real pieces are wild/non-measurable.
2. Where Choice Appears
Sketch. Build a paradoxical decomposition using a free subgroup of SO(3). Selecting one representative from each orbit requires an arbitrary global selector — a choice function. Without AC (or similar strength), the construction cannot proceed.
The pieces are not Lebesgue measurable, so measure additivity doesn’t constrain the result.
See Axiom of Choice for equivalents and fragments (DC, CC, Ultrafilter Lemma).
3. Dimensions & Amenability
- ℝ² (amenable): No Banach–Tarski; planar isometry group is amenable.
- ℝ³ and higher (non-amenable actions): BT and related paradoxes appear.
4. Avoiding BT
- Change axioms: Work in ZF or ZF+DC; in Solovay’s model (ZF + “every set of reals measurable”), BT fails.
- Restrict sets: Only allow measurable/Borel pieces → no paradoxical decompositions.
- Restrict groups: Limit to amenable group actions.
References
- Banach & Tarski (1924). Sur la décomposition des ensembles de points.
- Wagon, S. The Banach–Tarski Paradox.
- Jech, T. Set Theory.