Eventual Capture on a Measurable Cardinal

TL;DR

This work analyzes higher localization cardinals $\mathfrak{b}_\kappa(\in^*)$ and $\mathfrak{d}_\kappa(\in^*)$ at regular and large cardinals, with a focus on measurable $\kappa$. It introduces and develops localization forcing $\mathbb{LOC}_{h,\kappa}$ and its club variant $\mathbb{LOC}^{cl}_{h,\kappa}$, proving precise ZFC constraints at measurables (e.g., $\mathfrak{b}_\kappa(\in^*)=\kappa^+$ and $\mathfrak{d}_\kappa(\in^*)=2^\kappa$) and exploring optimality via forcing relative to supercompact cardinals. The authors establish consistency results separating these invariants from classical ones like $\mathfrak{s}_\kappa$ and $\mathfrak{p}_\kappa$, including cases where club variants differ from the standard, and show how ultrapower cofinalities and covering properties constrain what embeddings can preserve measurability. They further analyze the interplay between localization invariants and ultrapower cofinalities, highlighting when $cf^V(j_U(\kappa))$ must be large or can vary with different ultrafilters. Overall, the paper clarifies how large-cardinal hypotheses constrain or enable large gaps among high-cardinal invariants and provides forcing techniques to finely tune these values while preserving key large-cardinal features.

Abstract

We continue the study from \cite{BrendleFreidmanMontoya, vandervlugtlocalizationcardinals} of localization cardinals $\mfb_κ(\in^*)$ and $\mfd_κ(\in^*)$ and their variants at regular uncountable $κ$. We prove that if $κ$ 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.

Theorems & Definitions (119)

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