Banach–Tarski paradox

The Banach–Tarski paradox is a theorem that a solid ball can be partitioned into finitely many pieces and reassembled into two identical balls, using nonmeasurable sets and the axiom of choice.

Free group: A group whose elements have no relations beyond those required by the group axioms. Rotations about two axes generate the free-group structure used to rearrange the pieces.
Equidecomposability: A relation between sets that can be partitioned into finitely many pieces and matched by allowed transformations. The theorem asserts that one ball is equidecomposable with two copies under rigid motions.
Vitali set: A nonmeasurable subset of the real line constructed by selecting representatives modulo rational translation. It is a simpler example of choice producing a set without Lebesgue measure.
Principle of non-measurable sets: The fact that not every subset of Euclidean space can be assigned a translation-invariant, countably additive volume. The paradox makes this limitation unavoidable when arbitrary subsets are allowed.
Group action: A rule assigning transformations of a set to elements of a group while preserving group composition. Rotations act on points of the sphere to organize the construction.
Lebesgue measure: A measure that assigns lengths, areas, or volumes to a broad class of subsets of Euclidean space. Its finite additivity explains why measurable pieces could not double a ball's volume.
Tarski's theorem on non-paradoxical decompositions: A theorem characterizing when a group action permits paradoxical decompositions of sets. It distinguishes group actions that allow Banach–Tarski-type duplication from those that do not.
Solovay model: A model of set theory in which every set of reals is Lebesgue measurable, assuming an inaccessible cardinal. It shows that rejecting full choice can eliminate the relevant nonmeasurable subsets.
Hausdorff paradox: A decomposition of the sphere showing that rotation-invariant finite additivity cannot extend to all subsets. It provides the spherical decomposition underlying the ball construction.
Rigid motion: A distance-preserving transformation, such as a rotation or translation. Reassembly permits pieces to move by rigid motions without stretching them.

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