On Ultrapowers and Cohesive Ultrafilters

Tom Benhamou

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arXiv · 2410.06275

On Ultrapowers and Cohesive Ultrafilters

Tom Benhamou

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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.

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Tom Benhamou. 2024-10-08. On Ultrapowers and Cohesive Ultrafilters. https://arxiv.org/abs/2410.06275

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