# On the universality of the nonstationary ideal

## TL;DR

This paper explores the universality of the nonstationary ideal (NS) in set theory, extending Burke's theorem to broader classes of filters, and characterizing large cardinal properties and JF3nsson cardinals through NS projections.

## Contribution

It extends Burke's universality results of NS to -systems of filters, characterizes supercompactness via NS projections, and links -projections to large cardinal properties.

## Key findings

- -systems of filters do not encompass all set-generic embeddings.
- Supercompactness characterized by short extenders and NS projections.
- -projections of NS relate to -embeddings like I2 and I3.

## 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.