Diamonds on trees
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912905646047232 |
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| author | Guzmán, Osvaldo López-Callejas, Carlos |
| author_facet | Guzmán, Osvaldo López-Callejas, Carlos |
| contents | We generalize the diamond principle and its variants using the notion of stationarity in trees introduced by Brodsky in [Brodsky, A. M., A theory of stationary trees and the balanced Baumgartner--Hajnal--Todorcevic theorem for trees. The Bulletin of Symbolic Logic]. In particular, we show that if $T$ is a nonspecial $ω_1$-tree, then $\diamondsuit_T \implies \diamondsuit$, and if $T$ is a Suslin tree, then $\diamondsuit_T \iff \diamondsuit$. We also prove that $\diamondsuit^*$ implies $\diamondsuit_T$ (yielding the consistency of $\diamondsuit_T$) and establish the consistency of $\neg\diamondsuit^* + (\forall T\text{ nonspecial }ω_1\text{-tree }(\diamondsuit_T))$. Finally, we demonstrate that it is consistent with $\diamondsuit$ that there exists a nonspecial $ω_1$-tree with $(\neg\diamondsuit_T)$, introducing two forcing properties -- $σ(S)$-closed and strategically closed in models -- which are preserved under countable support iterations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diamonds on trees Guzmán, Osvaldo López-Callejas, Carlos Logic We generalize the diamond principle and its variants using the notion of stationarity in trees introduced by Brodsky in [Brodsky, A. M., A theory of stationary trees and the balanced Baumgartner--Hajnal--Todorcevic theorem for trees. The Bulletin of Symbolic Logic]. In particular, we show that if $T$ is a nonspecial $ω_1$-tree, then $\diamondsuit_T \implies \diamondsuit$, and if $T$ is a Suslin tree, then $\diamondsuit_T \iff \diamondsuit$. We also prove that $\diamondsuit^*$ implies $\diamondsuit_T$ (yielding the consistency of $\diamondsuit_T$) and establish the consistency of $\neg\diamondsuit^* + (\forall T\text{ nonspecial }ω_1\text{-tree }(\diamondsuit_T))$. Finally, we demonstrate that it is consistent with $\diamondsuit$ that there exists a nonspecial $ω_1$-tree with $(\neg\diamondsuit_T)$, introducing two forcing properties -- $σ(S)$-closed and strategically closed in models -- which are preserved under countable support iterations. |
| title | Diamonds on trees |
| topic | Logic |
| url | https://arxiv.org/abs/2511.12736 |