Algebra and Logic

, Volume 49, Issue 1, pp 91–98 | Cite as

A continuous version of the Hausdorff–Banach–Tarski paradox

  • V. A. Churkin

We come up with a simple proof for a continuous version of the Hausdorff–Banach–Tarski paradox, which does not make use of Robinson’s method of compatible congruences and fits in the case of finite and countable paradoxical decompositions. It is proved that there exists a free subgroup whose rank is of the power of the continuum in a rotation group of a three-dimensional Euclidean space. We also argue that unbounded subsets of Euclidean space containing inner points are denumerably equipollent.


Hausdorff–Banach–Tarski paradox continuous decompositions free subgroups of rotation group of Euclidean space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Hausdorff, Gesammelte Werke, Band IV: Analysis, Algebra und Zahlentheorie, Herausgegeben von S. D. Chatterji, R. Remmert und W. Scharlau, Springer-Verlag, Berlin (2001).Google Scholar
  2. 2.
    S. Banach and A. Tarski, “Sur la décomposition des ensembles de points en parties respectivement congruentes,” Fund. Math., 6, 244–277 (1924).Google Scholar
  3. 3.
    R. M. Robinson, “On the decomposition of spheres,” Fund. Math., 34, 246–260 (1947).zbMATHMathSciNetGoogle Scholar
  4. 4.
    J. Mycielski, “On the paradox of the sphere,” Fund. Math., 42, No. 2, 348–355 (1955).zbMATHMathSciNetGoogle Scholar
  5. 5.
    W. Sièrpiński, “Súr le paradoxe de la sphère,” Fund. Math., 33, 235–244 (1945).zbMATHMathSciNetGoogle Scholar
  6. 6.
    T. J. Dekker and J. de Groot, “Decompositions of a sphere,” Fund. Math., 43, No. 1, 185–194 (1956).zbMATHMathSciNetGoogle Scholar
  7. 7.
    S. Wagon, The Banach–Tarski Paradox, Encycl. Math. Its Appl., 24, Cambridge Univ. Press, Cambridge (1984).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations