Partitions of $\mathbb{R}^3$ into unit circles with no well-ordering of the reals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909448914599936 |
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| author | Fatalini, Azul |
| author_facet | Fatalini, Azul |
| contents | Using a well-ordering on the reals, one can prove there exists a partition of the three-dimensional Euclidean space into unit circles (PUC). We show that the converse does not hold: there exist models of $\mathsf{ZF}$ without a well-ordering of the reals in which such partition exists. Specifically, we prove that the Cohen model has a PUC and construct a model satisfying $\mathsf{DC}$ where this is also the case. Furthermore, we present a general framework for constructing similar models for other paradoxical sets, under some conditions of extendability and amalgamation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Partitions of $\mathbb{R}^3$ into unit circles with no well-ordering of the reals Fatalini, Azul Logic Metric Geometry Using a well-ordering on the reals, one can prove there exists a partition of the three-dimensional Euclidean space into unit circles (PUC). We show that the converse does not hold: there exist models of $\mathsf{ZF}$ without a well-ordering of the reals in which such partition exists. Specifically, we prove that the Cohen model has a PUC and construct a model satisfying $\mathsf{DC}$ where this is also the case. Furthermore, we present a general framework for constructing similar models for other paradoxical sets, under some conditions of extendability and amalgamation. |
| title | Partitions of $\mathbb{R}^3$ into unit circles with no well-ordering of the reals |
| topic | Logic Metric Geometry |
| url | https://arxiv.org/abs/2501.03131 |