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 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2010, Volume 49, Number 1, Pages 135–145 (Mi al431)

A continuous version of the Hausdorff–Banach–Tarski paradox

V. A. Churkinab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia

Abstract: 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.

Keywords: Hausdorff–Banach–Tarski paradox, continuous decompositions, free subgroups of rotation group of Euclidean space.

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English version:
Algebra and Logic, 2010, 49:1, 91–98

Bibliographic databases:

UDC: 512.543.12

Citation: V. A. Churkin, “A continuous version of the Hausdorff–Banach–Tarski paradox”, Algebra Logika, 49:1 (2010), 135–145

Citation in format AMSBIB
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\paper A continuous version of the Hausdorff--Banach--Tarski paradox
\jour Algebra Logika
\yr 2010
\vol 49
\issue 1
\pages 135--145
\mathnet{http://mi.mathnet.ru/al431}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2664125}
\zmath{https://zbmath.org/?q=an:1195.03044}
\transl
\jour Algebra and Logic
\yr 2010
\vol 49
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\pages 91--98
\crossref{http://dx.doi.org/10.1007/s10469-010-9080-y}
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