Parametrized Measuring and Club Guessing

TL;DR

This paper introduces Strong Measuring, a strengthened principle related to measuring closed sets on ω₁, and demonstrates its consistency with the negation of CH using forcing axioms, solving an open problem.

Contribution

It establishes the consistency of Strong Measuring with the negation of CH, extending the understanding of measuring principles in set theory.

Findings

Strong Measuring is a maximal strengthening of Moore's Measuring principle.

Strong Measuring is consistent with the negation of CH under certain axioms.

The paper connects Strong Measuring with MRP, Martin's Axiom, and BPFA.

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 CH is shown, solving an open problem from about parametrized measuring principles. Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $\sigma$-centered forcings, as well as from BPFA. We also consider strong versions of Measuring in the absence of the Axiom of Choice.