Compactness and Comparison

TL;DR

This paper explores the relationship between strongly compact and supercompact cardinals under the Ultrapower Axiom, showing that strongly compact cardinals are either supercompact or limits of supercompact cardinals, and analyzing ultrafilter factorizations.

Contribution

It characterizes strongly compact cardinals in terms of supercompactness using the Ultrapower Axiom and proves ultrafilter factorization results assuming GCH.

Findings

Strongly compact cardinals are either supercompact or limits of supercompact cardinals.

Under the Ultrapower Axiom and GCH, countably complete ultrafilters factor as finite iterations of supercompact ultrafilters.

The Ultrapower Axiom generalizes the linearity of the Mitchell order on normal ultrafilters.

Abstract

Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.