The Banach-Tarski paradox (Q2814161)

A bit more than thirty years ago the second author published the first edition of this book [Zbl 0569.43001]. In it he explained, among other things, one of the more fascinating results from (descriptive) set theory: one can decompose the (solid) unit ball in three-space into finitely many subsets and reassemble these finitely many sets into two copies of the unit ball, or one can decompose that same ball into finitely pieces and reassemble these pieces into a ball of double the original radius.NEWLINENEWLINEThis result, known as the `Banach-Tarski paradox' [\textit{St. Banach} and \textit{A. Tarski}, Fund. math. 6, 244--277 (1924; JFM 50.0370.02)], delights and fascinates many that first hear of it, and also abhors some. One can interpret this result in various ways: the sets in the partition are extreme\slash very nice examples of non-Lebesgue measurable sets, the axiom of choice has weird consequences, there is no (finitely additive) measure on the family of all subsets of~\(\mathbb{R}^3\) that is invariant under all isometries, \dotsNEWLINENEWLINEThis second edition largely follows the format of the first but with the addition of much new material. To begin there is of course the Banach-Tarski paradox itself, preceded by Hausdorff's paradoxical decomposition of the sphere on which it is based. This presentation should be accessible to an undergraduate student with an interest in set theory and algebra. After this the book explores what makes a paradoxical decomposition possible: the group of isometries of three-space contains the free product~\(\mathbb{Z}_2*\mathbb{Z}_3\) (consisting of rotations). That group has a paradoxical decomposition and this can be turned into a decomposition of the sphere by taking a choice set for the family of orbits under the action of~\(\mathbb{Z}_2*\mathbb{Z}_3\) on the sphere. Thus, one obtains paradoxical decompositions in the hyperbolic plane as well: its isometry group contains~\(\mathbb{Z}_2*\mathbb{Z}_3\) as well. Any time one has a group \(G\) acting on a set~\(X\) and \(G\)~contains~\(\mathbb{Z}_2*\mathbb{Z}_3\) then some sort of paradoxical decomposition is possible.NEWLINENEWLINEThe first part of the book finishes with \textit{M. Laczkovich}'s solution [J. Reine Angew. Math. 404, 77--117 (1990; Zbl 0748.51017)] of Tarski's circle-squaring problem. Since Banach had proved that there is a finitely additive extension of planar Lebesgue measure to the power set of the plane one knows that if one decomposes a measurable set into finitely many pieces and reassembles these into a new measurable set then the new set will have the same measure as the original one. Tarski asked whether one could decompose a disc of radius~\(1\), say, into finitely many pieces and reassembles these into a square of area~\(\pi\). Laczkovich showed that this is indeed possible; the construction uses many non-measurable pieces but it is applicable to many pairs of subsets of the plane.NEWLINENEWLINEThe second part of the book goes the other way and investigates when paradoxical decompositions are \textit{impossible}. Thus, we are treated to conditions that ensure that sets have measures invariant under various group actions.NEWLINENEWLINEOf note is also the very last chapter on the necessity of the axiom of choice in the constructions of the decompositions. It reports on some results obtained after publication of the first edition: in [Fundam. Math. 138, No. 1, 21--22 (1991; Zbl 0792.28006)] \textit{J. Pawlikowski} showed that the Banach-Tarski paradox can be proved using just the Hahn-Banach theorem.NEWLINENEWLINEOne of the main open problems posed in the present was solved by \textit{A. S. Marks} and \textit{S. T. Unger} in [``Borel circle squaring'', Preprint, \url{arXiv:1612.05833}]: one can solve Tarski's circle-squaring problem constructively, using Borel sets only.