The Weihrauch lattice at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$: the Cantor-Bendixson theorem
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| Format: | Preprint |
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2022
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| _version_ | 1866908443308195840 |
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| author | Cipriani, Vittorio Marcone, Alberto Valenti, Manlio |
| author_facet | Cipriani, Vittorio Marcone, Alberto Valenti, Manlio |
| contents | This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor-Bendixson theorem and various problems arising from them. In the framework of reverse mathematics these theorems are equivalent respectively to $\mathsf{ATR}_0$ and $\boldsymbolΠ_1^1\mathsf{-CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this is the first systematic study of problems at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$ in the Weihrauch lattice.
We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_15556 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Weihrauch lattice at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$: the Cantor-Bendixson theorem Cipriani, Vittorio Marcone, Alberto Valenti, Manlio Logic 03D78 (Primary) 03E15, 03B30, 03D30 (Secondary) This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor-Bendixson theorem and various problems arising from them. In the framework of reverse mathematics these theorems are equivalent respectively to $\mathsf{ATR}_0$ and $\boldsymbolΠ_1^1\mathsf{-CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this is the first systematic study of problems at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough. |
| title | The Weihrauch lattice at the level of $\boldsymbolΠ_1^1\mathsf{-CA}_0$: the Cantor-Bendixson theorem |
| topic | Logic 03D78 (Primary) 03E15, 03B30, 03D30 (Secondary) |
| url | https://arxiv.org/abs/2210.15556 |