Ranked Forcing and the Length of Generalized Borel Hierarchies
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917322150641664 |
|---|---|
| author | Chapman, Nick |
| author_facet | Chapman, Nick |
| contents | We extend A. Miller's framework of $α$-forcing to the case of a regular uncountable cardinal $κ= κ^{<κ}$ and apply it to study the structure of the $κ$-Borel hierarchy on subspaces of the generalized Baire space ${}^κκ$. We isolate a class of iterations of $α$-forcing and show that it satisfies a certain combinatorial property of admitting a sufficiently rich family of rank functions; this fact is then used to construct several models in which nontrivial constellations for the length of the $κ$-Borel hierarchy on multiple subspaces of ${}^κκ$ are realized simultaneously. Finally, we provide a higher variant of Steel's forcing with tagged trees and generalize arguments of Stern to derive the exact $κ$-Borel complexity of certain classes of well-founded trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07377 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ranked Forcing and the Length of Generalized Borel Hierarchies Chapman, Nick Logic 03E15 (Primary) 03E35, 54H05 (Secondary) We extend A. Miller's framework of $α$-forcing to the case of a regular uncountable cardinal $κ= κ^{<κ}$ and apply it to study the structure of the $κ$-Borel hierarchy on subspaces of the generalized Baire space ${}^κκ$. We isolate a class of iterations of $α$-forcing and show that it satisfies a certain combinatorial property of admitting a sufficiently rich family of rank functions; this fact is then used to construct several models in which nontrivial constellations for the length of the $κ$-Borel hierarchy on multiple subspaces of ${}^κκ$ are realized simultaneously. Finally, we provide a higher variant of Steel's forcing with tagged trees and generalize arguments of Stern to derive the exact $κ$-Borel complexity of certain classes of well-founded trees. |
| title | Ranked Forcing and the Length of Generalized Borel Hierarchies |
| topic | Logic 03E15 (Primary) 03E35, 54H05 (Secondary) |
| url | https://arxiv.org/abs/2603.07377 |