Ketonen's question and other cardinal sins

TL;DR

This paper proves that a weakly compact cardinal with an indecomposable ultrafilter need not be measurable, using a novel model-theoretic approach, and applies this technique to solve related open problems in set theory.

Contribution

It introduces a new proof technique analyzing decreasing sequences of ZFC models and applies it to settle multiple open questions in large cardinal and combinatorial set theory.

Findings

Weakly compact cardinals with indecomposable ultrafilters need not be measurable

The new proof technique is effective for solving open problems in set theory

Multiple open problems by Bagaria, Magidor, Lambie-Hanson, and Rinot are resolved

Abstract

Answering a question of Ketonen from the late 1970's, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing the limit of a decreasing sequence of models of ZFC. The utility of this proof technique is demonstrated further in this paper, where a problem by Bagaria and Magidor concerning strong compactness, and a problem by Lambie-Hanson and Rinot concerning the $C$-sequence number are solved as well.