Small models, large cardinals, and induced ideals

TL;DR

This paper establishes a framework linking large cardinal notions to filters on small models, enabling a canonical assignment of ideals that reflect the hierarchy and implications of large cardinal properties.

Contribution

It introduces a novel method to characterize large cardinals via filters on small models, unifying and extending classical large cardinal ideals.

Findings

Many large cardinal notions up to measurability are characterized by filters.

The assigned ideals match classical large cardinal ideals where previously defined.

Relations between these ideals mirror the hierarchy and implications of large cardinal properties.

Abstract

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct implication and consistency strength.