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Parametrized Measuring and Club Guessing

Volume 249 / 2020

David Asperó, John Krueger 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.

Authors

  • David AsperóSchool of Mathematics
    University of East Anglia
    Norwich NR4 7TJ, UK
    e-mail
  • John KruegerDepartment of Mathematics
    University of North Texas
    1155 Union Circle #311430
    Denton, TX 76203, U.S.A.
    e-mail

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