On the universality of the nonstationary ideal

Abstract

Burke \cite{MR1472122} proved that the generalized nonstationary ideal, denoted NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin-Keisler projection of the restriction of $\text{NS}$ to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of \emph{$\mathcal{C}$-systems of filters} introduced by Audrito-Steila \cite{AudritoSteila}. First we answer a question of \cite{AudritoSteila}, by proving that $\mathcal{C}$-systems of filters do not capture all kinds of set-generic embeddings. We provide a characterization of supercompactness in terms of short extenders and canonical projections of NS, without any reference to the strength of the extenders; as a corollary, NS can consistently fail to canonically project to arbitrarily strong short extenders. We prove that $\omega$-cofinal towers of normal ultrafilters---e.g.\ the kind used to characterize I2 and I3 embeddings---are well-founded if and only if they are canonical projections of NS. Finally, we provide a characterization of "$\aleph_\omega$ is Jonsson" in terms of canonical projections of NS.