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On nonparadoxical sets. (English) Zbl 0763.04005
Summary: Subsets $$A$$, $$B$$ of $$\mathbb{R}^ n$$ are countably equidecomposable if there is a partition $$\{A_ m: m\in\omega\}$$ of $$A$$ and isometries $$g_ m$$ of $$\mathbb{R}^ n$$ such that $$\{g_ m A_ m: m\in\omega\}$$ is a partition of $$B$$. A set $$A$$ is paradoxical if $$A$$ contains two disjoint subsets each countably equidecomposable with $$A$$. We show the existence of nonparadoxical sets of full Lebesgue measure. We also prove that every set of positive measure contains an uncountable paradoxial subset of full measure. A subset $$A$$ of $$\mathbb{R}^ n$$ is hereditarily nonparadoxical if $$A$$ has no uncountable paradoxical subsets. It is shown that the family of hereditarily nonparadoxical sets is a proper ideal and that, under $$\neg\text{CH}$$, the union of countably many hereditarily nonparadoxical sets has inner measure zero. This generalizes a result by Erdős and Kunen. We answer related questions concerning sets without repeated distances.

##### MSC:
 03E05 Other combinatorial set theory 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets
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