On Ultrapowers and Cohesive Ultrafilters

TL;DR

This paper characterizes the Tukey order and cohesive ultrafilters via ultrapowers, improves classical theorems, and explores spectra of ultrafilters, linking set-theoretic properties with large cardinal assumptions.

Contribution

It provides a new ultrapower-based characterization of ultrafilter orders, improves foundational theorems, and investigates ultrafilter spectra under large cardinal hypotheses.

Findings

Characterization of the Tukey order using ultrapowers.

Improved theorems of Kanamori from the 1970s.

Established conditions for the point and depth spectra of ultrafilters.

Abstract

We characterize the Tukey order, the Galvin property/ Cohesive ultrafilters from \cite{Kanamori1978} in terms of ultrapowers. We use this characterization to measure the distance between the Tukey order and other well-known orders of ultrafilters. Secondly, we improve two theorems of Kanamori \cite{Kanamori1978} from the 70's. We then study the point spectrum and the depth spectrum of an ultrafilter, and give a simple positive answer to Kanamori's question \cite[Question 2]{Kanamori1978} starting from a supercompact cardinal. We also prove that a positive answer requires more than $o(\kappa)=\kappa^{++}$. Finally, we prove several consistency results regarding the point and depth spectrum.