Multiple gaps and some finitizations of club and CH
Jorge Antonio Cruz Chapital
TL;DR
The paper investigates how capturing schemes over $oldsymbol{ω}_1$, mediated by CH and Ostaszewski's $ ext{clubsuit}$ principle, generate uncountable combinatorial and topological objects such as $oldsymbol{ω}_1$-gaps and entangled sets. It develops construction schemes and capturing axioms $CA^ ho$, $CA^ abla$ (and their variants), showing $CA^ ho$ follows from $ ext{varclubsuit}$ and $CA^ abla$ from CH, and proves a key result that $ ho$- or $ abla$-capturing schemes yield $oldsymbol{n}$-inseparable AD families coding every $oldsymbol{ω_1}$-like order; these methods produce numerous uncountable gaps and large gap cohomology groups. The forcing framework $ ext{P}(\mathcal{F})$ is employed to build schemes with preserving properties like IH$_1$ and IH$_ ho$, enabling recursive construction of uncountable objects. Additionally, CA$^ abla$ yields entangled sets and a metric space on $oldsymbol{ω_1}$ with no uncountable monotone subspaces, illustrating broad interactions between guessing principles, forcing, and the geometry of gaps. Overall, the work extends the toolbox for producing and analyzing uncountable combinatorial and topological structures under CH and related principles, connecting AD representations, construction schemes, and forcing in a unified framework.
Abstract
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.