Formal treatment
The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembly. Its mathematical structure is greatly elucidated by emphasizing the role played by the group of Euclidean motions and introducing the notions of equidecomposable sets and paradoxical set. Suppose that G is a group acting on a set X. In the most important special case, X is an n-dimensional Euclidean space, and G consists of all isometries of X, i.e. the transformations of X into itself that preserve the distances, usually denoted E(n). Two geometric figures that can be transformed into each other are called congruent, and this terminology will be extended to the general G-action. Two subsets A and B of X are called G-equidecomposable, or equidecomposable with respect to G, if A and B can be partitioned into the same finite number of respectively G-congruent pieces. This defines an equivalence relation among all subsets of X. Formally, if there exist non-empty sets A1, ..., Ak, B1, ..., Bk such that
{\displaystyle A=\bigcup _{i=1}^{k}A_{i},\quad B=\bigcup _{i=1}^{k}B_{i},}
{\displaystyle \quad A_{i}\cap A_{j}=\emptyset ,\quad B_{i}\cap B_{j}=\emptyset \quad {\text{for all }}1\leq i<j\leq k,}
and there exist elements {\displaystyle g_{i}\in G} such that
{\displaystyle g_{i}(A_{i})=B_{i}{\text{ for all }}1\leq i\leq k,}
then it can be said that A and B are G-equidecomposable using k pieces. If a set E has two disjoint subsets A and B such that A and E, as well as B and E, are G-equidecomposable then E is called paradoxical.