A continuous version of the Hausdorff–Banach–Tarski paradox |
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Authors: | V A Churkin |
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Institution: | 1.Sobolev Institute of Mathematics, Siberian Branch,Russian Academy of Sciences,Novosibirsk,Russia;2.Novosibirsk State University,Novosibirsk,Russia |
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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. |
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