Trevor Wilson, Miami University
The large cardinal strength of Vopenka's Principle for trees and for rayless trees
Vopenka's Principle (VP) says that for every proper class of structures with the same signature, there is an elementary embedding from one structure in the class to another. An equivalent form of VP says that for every proper class of graphs, there is an embedding from one graph in the class to another; let us denote this form by VP(graphs, embeddings) with the obvious meaning. We can obtain weaker instances of VP by restricting to particular kinds of graphs such as trees, which are connected acyclic graphs, and rayless trees, which are trees with no infinite path. We will show that VP(trees, embeddings) and VP(rayless trees, embeddings) occupy two different places in the large cardinal hierarchy below VP, and that each is equivalent to the existence of certain virtual large cardinals. Namely, we will show that VP(trees, embeddings) is equivalent to the existence of a weakly virtually A-extendible cardinal (as defined by Gitman and Hamkins) for every class A, and VP(rayless trees, embeddings) is equivalent to the existence of what we will call a weakly virtually A-strong cardinal for every class A. For a better-known point of comparison: the former large cardinal hypothesis is stronger than the existence of a remarkable cardinal, whereas the latter is weaker. We will also relate these two instances of VP to other variants of VP such as generic Vopenka's Principle (as defined by Bagaria, Gitman, and Schindler) and generic Weak Vopenka's Principle.