Banach–Tarski paradox

The Banach–Tarski paradox is a famous theorem in set-theoretic geometry. It describes a process that seems to defy the laws of physics. The theorem states that a solid ball in three-dimensional space can be divided into a finite number of disjoint subsets. These subsets can then be reassembled using only rotations and translations. Once reassembled, they form two identical copies of the original ball. This process does not involve stretching, bending, or adding any new points to the set.

To many, this sounds impossible because it seems to double the volume of the object. However, this is a veridical paradox. This means it is a mathematical truth that contradicts our physical intuition but contains no logical contradictions.

The mechanism of this paradox relies on the nature of the pieces used in the decomposition. These pieces are not solid chunks like you would find in a real-world object. Instead, they are non-measurable sets. In mathematics, a non-measurable set is a collection of points that does not have a defined volume. Because these subsets lack a measurable volume, the standard rules of volume preservation do not apply. You are not adding matter; you are simply rearranging infinite scatterings of points.

While the individual pieces have no defined volume, the final reassembled balls do have a defined volume. This explains why the total volume can change during the process.

There are different versions and scales of this theorem. The basic version focuses on doubling a single ball. A more powerful version is known as the strong form of the Banach–Tarski paradox. This version states that any two bounded subsets of three-dimensional Euclidean space are equidecomposable. This is true as long as both subsets have a non-empty interior. This leads to the famous informal description called the "pea and the Sun" paradox. It suggests that a tiny pea could be chopped up and reassembled into the Sun.

The history of this discovery is linked to several important mathematicians. Stefan Banach and Alfred Tarski published their construction in a paper in 1924. Their work built upon earlier mathematical foundations. Giuseppe Vitali created a similar construction regarding the unit interval in 1905. Felix Hausdorff also explored paradoxical decompositions of the sphere in 1914. These researchers paved the way for Banach and Tarski to explore more complex subsets of Euclidean spaces.

Specific mathematical constraints define how this paradox operates. For example, the number of pieces required is quite small. A researcher named Raphael M. Robinson proved that doubling a ball can be done with exactly five pieces. He also showed that it is impossible to achieve this with fewer than five pieces. Furthermore, the paradox is strictly dependent on the number of dimensions. It is false in one or two dimensions. It only works in three or more dimensions because of the richer structure of Euclidean motions in those spaces.

The proof of the paradox relies heavily on the axiom of choice. This is a critical rule in set theory that allows for the construction of non-measurable sets. The axiom of choice allows mathematicians to make an uncountable number of choices to build these complex sets. Without this axiom, the paradox cannot be proven. In fact, the paradox is not a theorem of standard Zermelo-Fraenkel set theory alone. It requires the addition of the axiom of choice to function.

This paradox has deep connections to other fields of mathematics. It helped motivate the study of amenable groups. A mathematician named John von Neumann studied the groups of equivalences that make these decompositions possible. He introduced the concept of amenability to describe groups where such paradoxes cannot exist. Tarski later proved that amenable groups are precisely those that do not allow for paradoxical decompositions. This research moved mathematics toward new directions in group theory and foundational logic.