Eventual Capture on a Measurable Cardinal

TL;DR

This paper investigates localization cardinals at measurable cardinals, showing they trivialize in that context, and establishes new consistency results and relations among these and other cardinal invariants.

Contribution

It proves that localization cardinals trivialize at measurable cardinals and constructs models with diverse values for these invariants, answering open questions.

Findings

Localization cardinals trivialize at measurable cardinals.

Constructs models with different values of _h(\u220a^*) and _h(^*) at measurable cardinals.

Establishes relations between localization cardinals and other cardinal invariants.

Abstract

We continue the study from \cite{BrendleFreidmanMontoya, vandervlugtlocalizationcardinals} of localization cardinals $\mfb_\kappa(\in^*)$ and $\mfd_\kappa(\in^*)$ and their variants at regular uncountable $\kappa$. We prove that if $\kappa$ is measurable then these cardinals trivialize. We also provide other fundamental restrictions in the most general setting. We prove the results are optimal by forcing different values for $\mathfrak{b}_{\id^+}(\in^*),\mathfrak{d}_{\id^{++}}(\in^*)$ at a measurable. As a by-product, we prove the consistency of $\mfb_h(\in^*) < \mfb_{h'}(\in^*)$ for functions $h, h' \in \kk$, thus answering a question of Brendle, Brooke-Taylor, Friedman and Montoya. Moreover, we study the relation between these cardinals and other well-known cardinal invariants.