Virtual Vopěnka's Principle
This is a talk at the Accessible categories and their connections Conference, University of Leeds, July 18, 2018.
Abstract: A major segment of modern research in set theory focuses on the topics of forcing and large cardinals. Forcing, introduced by Cohen to show the independence of the Continuum Hypothesis, is a technique for building set-theoretic universes satisfying a desired list of properties. Large cardinal axioms are strong axioms positing the existence of very large infinite objects and (often) elementary embeddings between set-theoretic structures. These axioms form a hierarchy against which the strength of any set-theoretic assertion can be measured. Forcing and large cardinals interact in a number of unexpected ways. In this talk, I will discuss a relatively new avenue of research involving the notion of virtual large cardinals. Given a large cardinal property $A$ characterized by the existence of elementary embeddings of first-order structures, we say that the property $A$ holds virtually if embeddings on structures from the universe $V$ characterizing $A$ exists in forcing extensions of $V$. Virtual large cardinals are much weaker than their original counterparts, for instance being compatible with the constructible universe $L$. This line of research leads naturally to considering a virtual version of Vopěnka's Principle, a large cardinal principle which has found applications in other branches of math. Virtual Vopěnka's Principle states that given a proper class of first-order structures in a common language, two of them must elementary embed in a forcing extension of $V$. I will discuss the consistency strength of the new principle, which like other virtual large cardinal principles, is compatible with $L$, and some open questions concerning stronger version of the principle, where we demand that the forcing extension in which the embeddings exist preserves a large initial segment of the universe. Parts of this work are joint with Bagaria, Schindler, and Hamkins.