Algebra and Logic

, Volume 49, Issue 1, pp 91–98

A continuous version of the Hausdorff–Banach–Tarski paradox


    • Sobolev Institute of Mathematics, Siberian BranchRussian Academy of Sciences
    • Novosibirsk State University

DOI: 10.1007/s10469-010-9080-y

Cite this article as:
Churkin, V.A. Algebra Logic (2010) 49: 91. doi:10.1007/s10469-010-9080-y

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

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© Springer Science+Business Media, Inc. 2010