Multiple gaps and some finitizations of club and CH
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912340348239872 |
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| author | Chapital, Jorge Antonio Cruz |
| author_facet | Chapital, Jorge Antonio Cruz |
| contents | We continue the development of the theory of capturing schemes over $ω_1$ by analyzing the relation between the capturing construction schemes (whose existence is implied by Jensen's $\Diamond$-principle) and both the Continuum Hypothesis and Ostaszewski's $\clubsuit$-principle. Formally, we show that the property of being capturing can be viewed as the conjunction of two properties, one of which is implied by $\clubsuit$ and the other one by CH. We apply these principles to construct multiple gaps, entangled sets and metric spaces without uncountable monotone subspaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple gaps and some finitizations of club and CH Chapital, Jorge Antonio Cruz Logic General Topology 03E02, 03E35, 03E65, 03E75 We continue the development of the theory of capturing schemes over $ω_1$ by analyzing the relation between the capturing construction schemes (whose existence is implied by Jensen's $\Diamond$-principle) and both the Continuum Hypothesis and Ostaszewski's $\clubsuit$-principle. Formally, we show that the property of being capturing can be viewed as the conjunction of two properties, one of which is implied by $\clubsuit$ and the other one by CH. We apply these principles to construct multiple gaps, entangled sets and metric spaces without uncountable monotone subspaces. |
| title | Multiple gaps and some finitizations of club and CH |
| topic | Logic General Topology 03E02, 03E35, 03E65, 03E75 |
| url | https://arxiv.org/abs/2504.15398 |