, Volume 49, Issue 1, pp 91-98
Date: 27 Mar 2010

A continuous version of the Hausdorff–Banach–Tarski paradox

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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.

*Supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools, grant NSh-344.2008.1.
Translated from Algebra i Logika, Vol. 49, No. 1, pp. 135-145, January-February, 2010.