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arXiv · 2504.15398

Multiple gaps and some finitizations of club and CH

Jorge Antonio Cruz Chapital
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Abstract

We continue the development of the theory of capturing schemes over $\omega_1$ 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.

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Jorge Antonio Cruz Chapital. 2025-04-21. Multiple gaps and some finitizations of club and CH. https://arxiv.org/abs/2504.15398

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