Well-founded Boolean ultrapowers as large cardinal embeddings

TL;DR

This paper explores well-founded Boolean ultrapowers, showing they produce large cardinal embeddings and unify forcing with large cardinal concepts in set theory.

Contribution

It introduces the concept of well-founded Boolean ultrapowers and demonstrates their role in connecting forcing and large cardinals.

Findings

Boolean ultrapowers can produce large cardinal embeddings

They unify forcing and large cardinal theories

The construction generalizes classical ultrapowers

Abstract

Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete Boolean algebra, rather than only on a power set algebra. When they are well-founded, the associated Boolean ultrapower embeddings exhibit a large cardinal nature, and the Boolean ultrapower construction thereby unifies two central themes of set theory---forcing and large cardinals---by revealing them to be two facets of a single underlying construction, the Boolean ultrapower.