Paradoxical Decompositions

Set-theoretic paradoxes of Hausdorff and Banach–Tarski
  • Joel H. Shapiro
Part of the Universitext book series (UTX)


In Chap.  10 we used the fixed-point theorem of Markov and Kakutani to show that every abelian group G is “amenable” in the sense that there is a G-invariant mean on the vector space B(G) of bounded, real-valued functions on G. We observed that existence of such a “mean” is equivalent to existence of a finitely additive probability “measure” on \(\mathcal{P}(G)\), the algebra of all subsets of G, and we asked if every group turns out to be amenable. We showed that the free group F2 on two generators is not amenable by finding within F2 four pairwise disjoint subsets that could be reassembled, using only group motions, into two copies of F2.

Now we’ll see how this “paradoxical” property of F2, along with the Axiom of Choice, leads to astonishing results in set theory, most notably the famous Banach–Tarski Paradox, often popularly phrased as: Each (three dimensional) ball can be partitioned into a finite collection of subsets which can then be reassembled, using only rigid motions, into two copies of itself. Even more striking: given two bounded subsets of \(\mathbb{R}^{3}\) with nonvoid interior, each can be partitioned into a finite collection of subsets that can be rigidly reassembled into the other. For this result the fixed-point theorem of Knaster and Tarski (Theorem  1.2) makes another appearance, this time to prove a far-reaching generalization of the Schröder–Bernstein Theorem.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Joel H. Shapiro
    • 1
  1. 1.Portland State UniversityPortlandUSA

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