Forcing Over a Free Suslin Tree
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929678905769984 |
|---|---|
| author | Krueger, John Stejskalova, Sarka |
| author_facet | Krueger, John Stejskalova, Sarka |
| contents | We introduce an abstract framework for forcing over a free Suslin tree with suborders of products of forcings which add some structure to the tree using countable approximations. The main ideas of this framework are consistency, separation, and the Key Property. We give three applications of this framework: specializing derived trees of a free Suslin tree, adding uncountable almost disjoint subtrees of a free Suslin tree, and adding almost disjoint automorphisms of a free Suslin tree. Using the automorphism forcing, we construct a model in which there is an almost Kurepa Suslin tree and a non-saturated Aronszajn tree, and there does not exist a Kurepa tree. This model solves open problems due to Bilaniuk, Moore, and Jin and Shelah. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Forcing Over a Free Suslin Tree Krueger, John Stejskalova, Sarka Logic 03E05, 03E35, 03E40 We introduce an abstract framework for forcing over a free Suslin tree with suborders of products of forcings which add some structure to the tree using countable approximations. The main ideas of this framework are consistency, separation, and the Key Property. We give three applications of this framework: specializing derived trees of a free Suslin tree, adding uncountable almost disjoint subtrees of a free Suslin tree, and adding almost disjoint automorphisms of a free Suslin tree. Using the automorphism forcing, we construct a model in which there is an almost Kurepa Suslin tree and a non-saturated Aronszajn tree, and there does not exist a Kurepa tree. This model solves open problems due to Bilaniuk, Moore, and Jin and Shelah. |
| title | Forcing Over a Free Suslin Tree |
| topic | Logic 03E05, 03E35, 03E40 |
| url | https://arxiv.org/abs/2402.00226 |