An almost Kurepa Suslin tree with strongly non-saturated square
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914026708008960 |
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| author | Krueger, John Mendoza, Eduardo Martinez |
| author_facet | Krueger, John Mendoza, Eduardo Martinez |
| contents | For uncountable downwards closed subtrees $U$ and $W$ of an $ω_1$-tree $T$, we say that $U$ and $W$ are strongly almost disjoint if their intersection is a finite union of countable chains. The tree $T$ is strongly non-saturated if there exists a strongly almost disjoint family of $ω_2$-many uncountable downwards closed subtrees of $T$. In this article we construct a Knaster forcing which adds a Suslin tree together with a family of $ω_2$-many strongly almost disjoint automorphisms of it (and thus the square of the Suslin tree is strongly non-saturated). To achieve this goal, we introduce a new idea called $ρ$-separation, which is an adaptation to the finite context of the notion of separation which was recently introduced by Stejskalová and the first author for the purpose of adding automorphisms of a tree with a forcing with countable conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10463 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An almost Kurepa Suslin tree with strongly non-saturated square Krueger, John Mendoza, Eduardo Martinez Logic 03E05, 03E35, 03E40 For uncountable downwards closed subtrees $U$ and $W$ of an $ω_1$-tree $T$, we say that $U$ and $W$ are strongly almost disjoint if their intersection is a finite union of countable chains. The tree $T$ is strongly non-saturated if there exists a strongly almost disjoint family of $ω_2$-many uncountable downwards closed subtrees of $T$. In this article we construct a Knaster forcing which adds a Suslin tree together with a family of $ω_2$-many strongly almost disjoint automorphisms of it (and thus the square of the Suslin tree is strongly non-saturated). To achieve this goal, we introduce a new idea called $ρ$-separation, which is an adaptation to the finite context of the notion of separation which was recently introduced by Stejskalová and the first author for the purpose of adding automorphisms of a tree with a forcing with countable conditions. |
| title | An almost Kurepa Suslin tree with strongly non-saturated square |
| topic | Logic 03E05, 03E35, 03E40 |
| url | https://arxiv.org/abs/2406.10463 |