Fr\'echet cardinals

TL;DR

This paper explores the concept of Fréchet cardinals, infinite cardinals where the Fréchet filter extends to a countably complete ultrafilter, and examines their relationship with strongly compact cardinals under the Ultrapower Axiom.

Contribution

It introduces the notion of Fréchet cardinals and investigates their connection to strongly compact cardinals assuming the Ultrapower Axiom.

Findings

Fréchet cardinals are characterized by the extension of the Fréchet filter to a countably complete ultrafilter.

Under the Ultrapower Axiom, a relationship between Fréchet and strongly compact cardinals is established.

The paper provides new insights into the structure of large cardinals and their interrelations.

Abstract

An infinite cardinal $\lambda$ is called Fr\'echet if the Fr\'echet filter on $\lambda$ extends to a countably complete ultrafilter. We investigate the relationship between Fr\'echet cardinals and strongly compact cardinals under a hypothesis called the Ultrapower Axiom.