Parametrized Measuring and Club Guessing
Volume 249 / 2020
Fundamenta Mathematicae 249 (2020), 169-183
MSC: Primary 03E05, 03E35; Secondary 03E57.
DOI: 10.4064/fm781-9-2019
Published online: 13 December 2019
Abstract
We introduce Strong Measuring, a maximal strengthening of J. T. Moore’s Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $\omega _1$ is measured by some club subset of $\omega _1$. The consistency of Strong Measuring with the negation of $\mathsf {CH}$ is shown, solving an open problem from Asperó and Mota’s 2017 preprint on Measuring. Specifically, we prove that Strong Measuring follows from $\mathsf {MRP}$ together with Martin’s Axiom for $\sigma $-centered forcings, as well as from $\mathsf {BPFA}$. We also consider strong versions of Measuring in the absence of the Axiom of Choice.
