Ketonen's question and other cardinal sins

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.