Generalized Borel Sets
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914164525498368 |
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| author | Agostini, Claudio Chapman, Nick Ros, Luca Motto Pitton, Beatrice |
| author_facet | Agostini, Claudio Chapman, Nick Ros, Luca Motto Pitton, Beatrice |
| contents | Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable cardinal $κ$, possibly singular, satisfying $2^{<κ}=κ$. We provide fundamental properties of the $κ^+$-Borel hierarchy of any regular Hausdorff space of weight at most $κ$, and establish sufficient conditions for its non-collapse. We highlight a unique phenomenon that arises in the case of singular cardinals, namely, the existence of a second, distinct Borel hierarchy, the $κ$-Borel hierarchy: we prove that it is strictly finer than the $κ^+$-Borel hierarchy, and then characterize the precise relationship between the two. Finally, for regular cardinals, we resolve three questions about the behavior of the $κ^+$-Borel hierarchy on subspaces of the generalized Baire space ${}^κκ$, constructing various models via forcing where several nontrivial constellations for the length of the $κ^+$-Borel hierarchy on the space are realized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15663 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Borel Sets Agostini, Claudio Chapman, Nick Ros, Luca Motto Pitton, Beatrice Logic 03E15 (Primary) 03E35, 03E47, 54H05, 54E99 (Secondary) Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable cardinal $κ$, possibly singular, satisfying $2^{<κ}=κ$. We provide fundamental properties of the $κ^+$-Borel hierarchy of any regular Hausdorff space of weight at most $κ$, and establish sufficient conditions for its non-collapse. We highlight a unique phenomenon that arises in the case of singular cardinals, namely, the existence of a second, distinct Borel hierarchy, the $κ$-Borel hierarchy: we prove that it is strictly finer than the $κ^+$-Borel hierarchy, and then characterize the precise relationship between the two. Finally, for regular cardinals, we resolve three questions about the behavior of the $κ^+$-Borel hierarchy on subspaces of the generalized Baire space ${}^κκ$, constructing various models via forcing where several nontrivial constellations for the length of the $κ^+$-Borel hierarchy on the space are realized. |
| title | Generalized Borel Sets |
| topic | Logic 03E15 (Primary) 03E35, 03E47, 54H05, 54E99 (Secondary) |
| url | https://arxiv.org/abs/2511.15663 |