Noah Schoem : An anti-saturation result for ideals

Posted by Filippo Calderoni , part of the Logic Seminar.

At

March 3, 2020, 3 p.m.

In

427 SEO

Abstract

An ideal $I$ on a set is said to be $\lambda$-saturated if no family of $I$-almost pairwise disjoint $I$-positive sets has cardinality $\lambda$. Sufficiently small saturation allows $I$ to generically create a nontrivial elementary embedding $j:V\to M$ in an outer model, serving as a kind of generic large cardinal axiom. We exhibit a new anti-saturation result. Inspired by a result of Cox and Eskew, we show that it is possible to force to destroy the saturation of ideals at an inaccessible cardinal, while preserving many of their large cardinal-type properties.