*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**.

Using this terminology, the Banach–Tarski paradox can be reformulated as follows:

- A three-dimensional Euclidean ball is equidecomposable with two copies of itself.

The strong version of the paradox claims:

- Any two bounded subsets of 3-dimensional Euclidean space with non-empty interiors are equidecomposable.

*A*is equidecomposable with a subset of

*B*and

*B*is equidecomposable with a subset of

*A*, then

*A*and

*B*are equidecomposable.

The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a bijective function that can map the points in one shape into the other in a one-to-one fashion. In the language of Georg Cantor's set theory, these two sets have equal cardinality. Thus, if one enlarges the group to allow arbitrary bijections of

*X*then all sets with non-empty interior become congruent. Likewise, we can make one ball into a larger or smaller ball by stretching, in other words, by applying similarity transformations. Hence if the group

*G*is large enough, we may find

*G*-equidecomposable sets whose "size" varies. Moreover, since a countable set can be made into two copies of itself, one might expect that somehow, using countably many pieces could do the trick.

On the other hand, in the Banach–Tarski paradox the number of pieces is finite and the allowed equivalences are Euclidean congruences, which preserve the volumes. Yet, somehow, they end up doubling the volume of the ball! While this is certainly surprising, some of the pieces used in the paradoxical decomposition are non-measurable sets, so the notion of volume (more precisely, Lebesgue measure) is not defined for them, and the partitioning cannot be accomplished in a practical way. In fact, the Banach–Tarski paradox demonstrates that it is impossible to find a finitely-additive measure (or a Banach measure) defined on all subsets of a Euclidean space of three (and greater) dimensions that is invariant with respect to Euclidean motions and takes the value one on a unit cube. In his later work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence of a finitely-additive invariant measure.

The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a

*F*

_{2}-paradoxical decomposition of

*F*

_{2}, the free group with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets

*B*,

*C*,

*D*and a countable set

*E*such that, on the one hand,

*B*,

*C*,

*D*are pairwise congruent, and, on the other hand,

*B*is congruent with the union of

*C*and

*D*. This is often called the

**Hausdorff paradox**.

- Find a paradoxical decomposition of the free group in two generators.
- Find a group of rotations in 3-d space isomorphic to the free group in two generators.
- Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
- Extend this decomposition of the sphere to a decomposition of the solid unit ball.

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