The Banach-tarski Paradox

The Banach-tarski Paradox

Stefan Banach and Alfred Tarski introduced the phrase: “a pea can be chopped up and reassembled into the Sun,” a seemingly impossible concept. Using this theorem as motivation, this paper will explore the existence of non-measurable sets and paradoxical decompositions as well as provide a sketch of the proof of the paradox.

The Banach-Tarski Paradox

Regular probability comparisons imply the Banach-Tarski Paradox

Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: (a) there can be nonmeasurable events that have no probability at all and (b) on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the Axiom of Choice (AC). We shall show that the existence of a...

The Banach-tarski Paradox

The Banach-Tarski paradox is one of the most celebrated paradoxes in mathematics. It states that given any two subsets A and B of R, which are bounded and have non-empty interior, it is possible to ‘cut’ A into a finite number of pieces which can be moved by rigid motions (translations and rotations) to form exactly B. This has many amusing consequences; it is most commonly stated as the assert...

Banach-Tarski paradox using pieces with the property of Baire.

In 1924 Banach and Tarski, using ideas of Hausdorff, proved that there is a partition of the unit sphere S2 into sets A1,...,Ak,B1,..., Bl and a collection of isometries [sigma1,..., sigmak, rho1,..., rhol] so that [sigma1A1,..., sigmakAk] and [rho1B1,..., rholBl] both are partitions of S2. The sets in these partitions are constructed by using the axiom of choice and cannot all be Lebesgue meas...