The Banach–Tarski Paradox (book)

The Banach–Tarski paradox, proved by Stefan Banach and Alfred Tarski in 1924, states that it is possible to partition a three-dimensional unit ball into finitely many pieces and reassemble them into two unit balls, a single ball of larger or smaller area, or any other bounded set with a non-empty interior. Although it is a mathematical theorem, it is called a paradox because it is so counter-intuitive; in the preface to the book, Jan Mycielski calls it the most surprising result in mathematics. It is closely related to measure theory and the non-existence of a measure on all subsets of three-dimensional space, invariant under all congruences of space, and to the theory of paradoxical sets in free groups and the representation of these groups by three-dimensional rotations, used in the proof of the paradox. The topic of the book is the Banach–Tarski paradox, its proof, and the many related results that have since become known.[3][5]

The book is divided into two parts, the first on the existence of paradoxical decompositions and the second on conditions that prevent their existence.[1][7] After two chapters of background material, the first part proves the Banach–Tarski paradox itself, considers higher-dimensional spaces and non-Euclidean geometry, studies the number of pieces necessary for a paradoxical decomposition, and finds analogous results to the Banach–Tarski paradox for one- and two-dimensional sets. The second part includes a related theorem of Tarski that congruence-invariant finitely-additive measures prevent the existence of paradoxical decompositions, a theorem that Lebesgue measure is the only such measure on the Lebesgue measurable sets, material on amenable groups, connections to the axiom of choice and the Hahn–Banach theorem.[3][7] Three appendices describe Euclidean groups, Jordan measure, and a collection of open problems.[1]

The second edition adds material on several recent results in this area, in many cases inspired by the first edition of the book. Trevor Wilson proved the existence of a continuous motion from the one-ball assembly to the two-ball assembly, keeping the sets of the partition disjoint at all times; this question had been posed by de Groot in the first edition of the book.[7][9] Miklós Laczkovich solved Tarski's circle-squaring problem, asking for a dissection of a disk to a square of the same area, in 1990.[7][8][10] And Edward Marczewski had asked in 1930 whether the Banach–Tarski paradox could be achieved using only Baire sets; a positive answer was found in 1994 by Randall Dougherty and Matthew Foreman.[8][11]

The book is written at a level accessible to mathematics graduate students, but provides a survey of research in this area that should also be useful to more advanced researchers.[3] The beginning parts of the book, including its proof of the Banach–Tarski paradox, should also be readable by undergraduate mathematicians.[4]

Reviewer Włodzimierz Bzyl writes that "this beautiful book is written with care and is certainly worth reading".[2] Reviewer John J. Watkins writes that the first edition of the book "became the classic text on paradoxical mathematics" and that the second edition "exceeds any possible expectation I might have had for expanding a book I already deeply treasured".[8]

• List of paradoxes
• History of mathematics