A continuous version of the Hausdorff-Banach-Tarski paradox

作者：Churkin V A^*

来源：Algebra and Logic, 2010, 49(1): 91-98.

DOI：10.1007/s10469-010-9080-y

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

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.

• 出版日期2010-3