Banach–Tarski paradox

Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their original shape.

The theorem is a veridical paradox: it contradicts basic geometric intuition, but is not false or self-contradictory.

"Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume.

Unlike most theorems in geometry, the mathematical proof of this result depends on the choice of axioms for set theory in a critical way.

[3] As proved independently by Leroy[4] and Simpson,[5] the Banach–Tarski paradox does not violate volumes if one works with locales rather than topological spaces.

Allowing for this hidden mass to be taken into account, the theory of locales permits all subsets (and even all sublocales) of the Euclidean space to be satisfactorily measured.

Then the proposition means that the original ball A can be divided into a certain number of pieces and then be rotated and translated in such a way that the result is the whole set B, which contains two copies of A.

The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed that an analogous statement remains true if countably many subsets are allowed.

He also found a form of the paradox in the plane which uses area-preserving affine transformations in place of the usual congruences.

Since only free subgroups are needed in the Banach–Tarski paradox, this led to the long-standing von Neumann conjecture, which was disproved in 1980.

In the most important special case, X is an n-dimensional Euclidean space (for integral n), 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.

Thus, if one enlarges the group to allow arbitrary bijections of X, then all sets with non-empty interior become congruent.

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.

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.

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.

Thus Banach and Tarski imply that AC should not be rejected solely because it produces a paradoxical decomposition, for such an argument also undermines proofs of geometrically intuitive statements.

However, in 1949, A. P. Morse showed that the statement about Euclidean polygons can be proved in ZF set theory and thus does not require the axiom of choice.

In 1991, using then-recent results by Matthew Foreman and Friedrich Wehrung,[9] Janusz Pawlikowski proved that the Banach–Tarski paradox follows from ZF plus the Hahn–Banach theorem.

[10] The Hahn–Banach theorem does not rely on the full axiom of choice but can be proved using a weaker version of AC called the ultrafilter lemma.

(if the ratio between the two angles is rational) or the free abelian group over two elements; either way, it does not have the property required in step 1.

The axiom of choice can be used to pick exactly one point from every orbit; collect these points into a set M. The action of H on a given orbit is free and transitive and so each orbit can be identified with H. In other words, every point in S2 can be reached in exactly one way by applying the proper rotation from H to the proper element from M. Because of this, the paradoxical decomposition of H yields a paradoxical decomposition of S2 into four pieces A1, A2, A3, A4 as follows: where we define and likewise for the other sets, and where we define (The five "paradoxical" parts of F2 were not used directly, as they would leave M as an extra piece after doubling, owing to the presence of the singleton {e}.)

The (majority of the) sphere has now been divided into four sets (each one dense on the sphere), and when two of these are rotated, the result is double of what was had before: Finally, connect every point on S2 with a half-open segment to the origin; the paradoxical decomposition of S2 then yields a paradoxical decomposition of the solid unit ball minus the point at the ball's center.

Using the fact that the free group F2 of rank 2 admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere Sn−1 can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the Sn−1 using rotations.

A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by John von Neumann: unlike the group SO(3) of rotations in three dimensions, the group E(2) of Euclidean motions of the plane is solvable, which implies the existence of a finitely-additive measure on E(2) and R2 which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets.

Von Neumann then posed the following question: can such a paradoxical decomposition be constructed if one allows a larger group of equivalences?

Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable.