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The Banach-Tarski paradox. (English) Zbl 0569.43001
Encyclopedia of Mathematics and Its applications, Vol. 24. Cambridge etc.: Cambridge University Press. XVI, 251 p. £25.00; \$ 37.50 (1985).
One of the most striking paradoxes in mathematics is the so-called Banach-Tarski paradox. It states that if A and B are two bounded subsets in $${\mathbb{R}}^ 3$$ each having non-empty interior, then A can be decomposed into finitely many pieces that can be rearranged using rigid motions to form B. This book is devoted to a detailed exposition of the Banach-Tarski paradox and all its consequences. It starts with a discussion of an earlier result of Hausdorff concerning non-measurable decompositions of a sphere showing that modulo a countable set ”half” of a sphere could be congruent to a ”third” of a sphere. Then the author proceeds to discuss the Banach-Tarski paradox and its many consequences for measure theory, and its relation to group theory and logic.
The book is divided into two parts. Part I deals with the theory of paradoxical decompositions or the non-existence of finitely additive measures. The Hausdorff paradox and the Banach-Tarski paradox are explained and treated by means of the mose elementary techniques that requires only undergraduate mathematics.
Part II is directed to the theory of finitely additive invariant measures or the non-existence of paradoxical decompositions. It finishes with a very interesting account of the role of the axiom of choice and the foundational implications of the Banach-Tarski paradox. In this section the author points out that the Banach-Tarski paradox is more important to mathematics than to foundational questions as it has led to the introduction of some new and very valuable mathematical ideas such as amenability in groups.
The book contains three appendices, one on Euclidean transformation groups, one on Jordan measure, and the last appendix lists a very interesting collection of unsolved problems. In addition to a list of symbols and a very extensive index it contains a complete set of references listing more than 260 papers dealing with the subject matter of the book.
I consider this book to be a most welcome and valuable addition to the mathematical literature. The author is to be congratulated for his effort.
Reviewer: W.A.J.Luxemburg

##### MSC:
 43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis 03E25 Axiom of choice and related propositions 28A75 Length, area, volume, other geometric measure theory 43A07 Means on groups, semigroups, etc.; amenable groups