Forcing Axioms and construction schemes
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908514631286784 |
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| author | Chapital, Jorge Antonio Cruz Guzman, Osvaldo Todorcevic, Stevo |
| author_facet | Chapital, Jorge Antonio Cruz Guzman, Osvaldo Todorcevic, Stevo |
| contents | We continue the development of the theory of construction schemes over $ω_1$ as introduced by the third author by studying their relation with forcing axioms. Formally, we introduce the cardinals $\mathfrak{m}^n_{\mathcal{F}}$ and use the consistency of $\mathfrak{m}^2_\mathcal{F}>ω_1$ to prove a fundamental result relating gaps and almost disjoint families over $ω$. The cardinals $\mathfrak{m}_\mathcal{F}$ are also used to prove some limiting results for contstruction schemes, some of which answer questions from \cite{schemescruz}. Finally, we show that PID implies the non-existence of $2$-capturing schemes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Forcing Axioms and construction schemes Chapital, Jorge Antonio Cruz Guzman, Osvaldo Todorcevic, Stevo Logic 03E35, 03E65, 03E75, 03E02 We continue the development of the theory of construction schemes over $ω_1$ as introduced by the third author by studying their relation with forcing axioms. Formally, we introduce the cardinals $\mathfrak{m}^n_{\mathcal{F}}$ and use the consistency of $\mathfrak{m}^2_\mathcal{F}>ω_1$ to prove a fundamental result relating gaps and almost disjoint families over $ω$. The cardinals $\mathfrak{m}_\mathcal{F}$ are also used to prove some limiting results for contstruction schemes, some of which answer questions from \cite{schemescruz}. Finally, we show that PID implies the non-existence of $2$-capturing schemes. |
| title | Forcing Axioms and construction schemes |
| topic | Logic 03E35, 03E65, 03E75, 03E02 |
| url | https://arxiv.org/abs/2509.01712 |