In this final chapter, we give a more detailed account of the role played by the Axiom of Choice (AC) in the theory of paradoxical decompositions. Ever since its discovery, the Banach-Tarski Paradox has caused many mathematicians to look critically at the Axiom of Choice. Indeed, as soon as the Hausdorff Paradox was discovered it was challenged because of its use of AC; E. Borel [21, p. 256] objected because the choice set was not explicitly defined. We shall discuss these criticisms in more detail later in this chapter, but first we deal with several technical points that are essential to understanding the connection between AC and the Banach-Tarski Paradox.

Results of modern set theory can be used to show that AC is indeed necessary to obtain the Banach-Tarski Paradox, in the sense that the paradox is not a theorem of ZF alone. Before we can explain why this is so we need to introduce some notation and discuss some technical points of set theory. If T is a collection of sentences in the language of set theory, for example, T = ZF or T = ZF + AC, then Con(T) is the assertion, also a statement of set theory in fact, that T is consistent, that is, that a contradiction cannot be derived from T using the usual methods of proof. We take Con(ZF) as an underlying assumption in all that follows. Gödel proved in 1938 that Con(ZF) implies (and so is equivalent to) Con(ZF + AC); thus AC does not contradict ZF (see [98, 99]).

This book is motivated by the following theorem of Hausdorff, Banach, and Tarski: Given any two bounded sets A and B in three-dimensional space R3, each having nonempty interior, one can partition A into finitely many disjoint parts and rearrange them by rigid motions to form B. This, I believe, is the most surprising result of theoretical mathematics. It shows the imaginary character of the unrestricted idea of a set in R3. It precludes the existence of finitely additive, congruence-invariant measures over all bounded subsets of R3 and it shows the necessity of more restricted constructions such as Lebesgue's measure.

In the 1950s, the years of my mathematical education in Poland, this result was often discussed. J. F. Adams, R. M. Robinson, and W. Sierpiński wrote about it; my Ph. D. thesis was motivated by it. (All this is referenced in this monograph.) Thus it is a great pleasure to introduce you to this book, where this striking theorem and many related results in geometry and measure theory, and the underlying tools of group theory, are presented with care and enthusiasm. The reader will also find some applications of the most recent advances of group theory to measure theory — work of Gromov, Margulis, Rosenblatt, Sullivan, Tits, and others.

But to me the interest of mathematics lies no more in its theorems and theories than in the challenge of its surprising problems.

Certain proofs and theorems involving equidecomposability would be much simplified if we could add sets. For instance, if X could literally be added to X to form 2X, then the fact that X is paradoxical could be stated simply as X = 2X. In fact, this can be done if we expand the group action appropriately so that multiple copies of X can be formed. This new context for discussing equidecomposability will allow us to state and prove theorems that otherwise would be very cumbersome. One of these is a cancellation law for equidecomposability that has several uses, the most important of which is its use in Tarski's theorem (9.2) relating paradoxical decompositions and invariant measures. Another application will be a proof that any two subsets of S2 with nonempty interior are equidecomposable using rotations. This expanded context for equidecomposability will also yield a simpler proof of Theorem 4.5 that a locally commutative action of a free non-Abelian group is paradoxical.

Definition 8.1. Suppose the group G acts on X. Define an enlarged action as follows. Let X* = X × N and let G* = {(g, π): g ∈ G and π is a permutation of N}, and let the group G* act on X* by (g, π)(x, n) = (g(x), π(n)). If A ⊆ X*, then those n ∈ N such that A has at least one element with second coordinate n are called the levels of A.

In Part I we saw that the main idea in the construction of a paradoxical decomposition of a set was to first get such a decomposition in a group acting on the set, and then transfer it to the set. An almost identical theme pervades the construction of invariant measures on a set X acted upon by a group G. If there is a finitely additive, left-invariant measure defined on all subsets of G, then it can be used to produce a finitely additive, G-invariant measure defined on all subsets of X. Such measures on X yield that X (and certain subsets of X) are not G-paradoxical.

It was von Neumann [246] who realized that such a transference of measures was possible, and he began the job of classifying the groups that bear measures of this sort. In this chapter we first study some properties of the class of groups having measures and show that it is fairly extensive, containing all solvable groups. We then give the important application to the case of isometries acting on the line or plane, obtaining the nonexistence of Banach-Tarski-type paradoxes in these two dimensions.

Definition 10.1. If, for a group G, µ is a finitely additive measure on P(G) such that µ(G) = 1 and µ is left-invariant (µ(gA) = µ(A) for g ∈ G, A ⊆ G), then µ will be called simply a measure on G.

If one analyzed carefully the proof presented in Chapter 3 that a sphere may be duplicated using rotations, one would find that that proof used ten pieces. More precisely, S2 = A ∪ B where A ∩ B = ∅ and A ∼4S2 and B ∼6S2. (By 3.6, 2.6, and 1.10, S2\D splits into A′ and B′ with A′ ∼2S2\D ∼3B′; 3.9 shows that S2 ∼2S2\D, whence A′ and B′ yield A, B ⊆ S2 with the properties claimed.) It is easy to see that at least four pieces are necessary whenever a set X, acted upon by a group G, is G-paradoxical. For if X contains disjoint A, B with A ∼mX ∼nB and m + n < 4, then one of m or n equals 1. If, say, m = 1, then X = g(A) for some g ∈ G whence A = g-1(X) = X and B = ∅, a contradiction. It turns out that an interesting feature of the rotation group's action on the sphere allows the minimal number of pieces to be realized: there are disjoint sets A, B ⊆ S2 such that A ∼2S2 ∼2B. Moreover, the techniques used to cut the number of pieces to a minimum lead to significant new ideas on how to deal with the fixed points of an action of a free group, adding to our ability to recognize when a group's action is paradoxical.

The following is a list of unsolved problems in the area of paradoxical decompositions, equidecomposability, and finitely additive measures. The order represents the author's view as to their interest and importance.

Marczewski's Problem, Circa 1930 (3.12, 9.9)

Is there a finitely additive, isometry-invariant measure on the Borel sets in Sn, n ≥ 2 (or Rn, n ≥ 3), that has total measure one (or, normalizes the unit cube) and vanishes on meager sets? Such a measure cannot be countably additive (9.15), and so by 13.5, it is consistent with ZF + DC that no such finitely additive Borel measure exists. An equivalent problem in the case of R3 is: Is the unit cube paradoxical using pieces that have the Property of Baire?

Tarski's Circle-Squaring Problem, 1924 (7.5)

Is a circle (with interior) in the plane equidecomposable to a square (necessarily of the same area)?

Variations

1. (a) (p. 102) Can a negative solution be obtained if the pieces are restricted to the Borel sets? It is known that restriction to pieces that are parts of Jordan curves or two-cells (interior of a Jordan curve) yields a negative solution.

2. (b) (3.14) Is a regular tetrahedron in R3 equidecomposable to a cube using measurable pieces? A restriction to polyhedral pieces yields a negative solution (Hilbert's Third Problem).

3. […]

While many properties of infinite sets and their subsets were considered to be paradoxical when they were discovered, the development of paradoxical decompositions really began with the formalization of measure theory at the beginning of the twentieth century. The now classic example (due to Vitali in 1905) of a non-Lebesgue measurable set was the first instance of the use of a paradoxical decomposition to show the nonexistence of a certain type of measure. Ten years later, Hausdorff constructed a truly surprising paradox on the surface of the sphere (again, to show the nonexistence of a measure), and this inspired some important work in the 1920s. Namely, there was Banach's construction of invariant measures on the line and in the plane (which required the discovery of the main ideas of the Hahn-Banach Theorem) and the famous Banach-Tarski Paradox on duplicating, or enlarging, spheres and balls. This latter result, which at first seems patently impossible, is often stated as: It is possible to cut up a pea into finitely many pieces that can be rearranged to form a ball the size of the sun!

Their construction has turned out to be much more than a curiosity. Ideas arising from the Banach-Tarski Paradox have become the foundation of a theory of finitely additive measures, a theory that involves much interplay between analysis (measure theory and linear functionals), algebra (combinatorial group theory), geometry (isometry groups), and topology (locally compact topological groups).

It has been known since antiquity that the notion of infinity leads very quickly to seemingly paradoxical constructions, many of which seem to change the size of objects by operations that appear to preserve size. In a famous example, Galileo observed that the set of positive integers can be put into a one-one correspondence with the set of square integers, even though the set of nonsquares, and hence the set of all integers, seems more numerous than the squares. He deduced from this that “the attributes ‘equal,’ ‘greater’ and ‘less’ are not applicable to infinite … quantities,” anticipating developments in the twentieth century, when paradoxes of this sort were used to prove the nonexistence of certain measures.

An important feature of Galileo's observation is its resemblance to a duplicating machine; his construction shows how, starting with the positive integers, one can produce two sets, each of which has the same size as the set of positive integers. The idea of duplication inherent in this example will be the main object of study in this book. The reason that this concept is so fascinating is that, soon after paradoxes such as Galileo's were being clarified by Cantor's theory of cardinality, it was discovered that even more bizarre duplications could be produced using rigid motions, which are distance-preserving (and hence also area-preserving) transformations. I refer to the Banach-Tarski Paradox on duplicating spheres or balls, which is often stated in the fanciful form: a pea may be taken apart into finitely many pieces that may be rearranged using rotations and translations to form a ball the size of the sun.

The early 1980s seemed like an ideal time for a book on the Banach-Tarski Paradox, in part because of several fascinating unsolved problems, some of which had resisted attacks for more than fifty years. The attacks continued, with amazingly fruitful results in the 1980s and 1990s. Who would have predicted that, within a few years, the two central unsolved problems in the area – the Marczewski problem on whether the pieces in the paradox can have the Property of Baire and Tarski's notorious question on whether a disk and a square can be equidecomposable, would both be solved? Moreover each of the questions was solved in a surprising manner: Dougherty and Foreman showed that a Banach-Tarski Paradox using only Borel and meager sets is possible; and Laczkovich showed that a disk can be squared using translations alone.

The present volume is little changed from the first version. An Addendum summarizes some of the developments of the 1980s. The reader who is interested in pursuing up-to-date results, and finding out about current questions of interest is urged to consult the excellent survey paper by Laczkovich cited below along with some other important papers in the area.

The types of transformations that are used to produce paradoxes in Euclidean spaces and on spheres are usually the Euclidean isometries, but occasionally more general affine maps arise. Since the affine group is useful in studying and classifying isometries, we summarize the relevant facts about affine transformations. The book by Hausner [92] is a good reference for a more detailed presentation.

Definition A.1. A bijection f: Rn → Rnis called affine if for all P, Q ∈ Rnand reals α, β with α + β = 1, f(αP + βQ) = αf(P) + βf(Q). The affine transformations ofRnform a group, which is denoted by An.

Geometrically, a bijection is affine if and only if it carries lines to lines and preserves the ratio of distances along a line. Any nonsingular linear transformation is affine, since a linear transformation satisfies Definition A.1 for all α, β, not just pairs summing to one. The group of nonsingular linear transformations of Rn is denoted by GLn (general linear group). Linear maps leave the origin fixed, but affine maps need not do so; all translations of Rn are affine. Let Tn denote the group of translations of Rn. Tn is isomorphic to the additive group of Rn because composition of translations corresponds to addition of the translation vectors. It is an extremely useful fact that every affine map has a canonical representation in terms of linear maps and translations.

The idea of cutting a figure into pieces and rearranging them to form another figure goes back at least to Greek geometry, where this method was used to derive area formulas for regions such as parallelograms. When forming such rearrangements, one totally ignores the boundaries of the pieces. The consideration of a notion of dissection in which every single point is taken into account, that is, a set-theoretic generalization of the classical geometric definition, leads to an interesting, and very general, equivalence relation. By studying the abstract properties of this new relation, Banach and Tarski were able to improve on Hausdorf's Paradox (2.3) by eliminating the need to exclude a countable subset of the sphere. Since geometric rearrangements will be useful too, we start with the classical definition in the plane.

Definition 3.1. Two polygons in the plane are congruent by dissection if one of them can be decomposed into finitely many polygonal pieces that can be rearranged using isometries (and ignoring boundaries) to form the other polygon.

It is clear that polygons that are congruent by dissection have the same area. The converse was proved in the early nineteenth century, and a simple proof can be given by efficiently making use of the fact that congruence by dissection is an equivalence relation (transitivity is easily proved by superposition, using the fact that the intersection of two polygons is a polygon (see Boltianskii [17, p. 50] or Eves [70, p. 233])).