Indecomposability of ultrafilters was introduced by Keisler as a natural weakening of the concept of measurability. The property was first studied in depth by Chudnovsky and Chudnovsky, Prikry, and Silver (see [4] and [5]). One intriguing result of this early work was the following theorem: if κ is an inaccessible cardinal and there is an indecomposable ultrafilter over κ, then κ is in fact ω-Mahlo. Silver asked whether this result could be strengthened to say that an inaccessible cardinal carrying an indecomposable ultrafilter must be measurable. We prove in this paper that this is not the case; we construct a model where κ is inaccessible and carries an indecomposable ultrafilter but κ is not even weakly compact.

The results in this paper form part of the research for my doctoral dissertation at the University of California, Berkeley. I would like to thank my thesis advisor, Robert Solovay, for his patient guidance during this endeavor.