The large cardinal strength of Weak Vop\v{e}nka's Principle

TL;DR

This paper establishes the equivalence between Weak Vopěnka's Principle and the large cardinal principle Ord is Woodin, linking a categorical statement to a significant set-theoretic large cardinal axiom.

Contribution

It proves the equivalence of Weak Vopěnka's Principle with the large cardinal principle Ord is Woodin, and refines the known large cardinal lower bound from measurable to C-strong cardinals.

Findings

Weak Vopěnka's Principle is equivalent to Ord is Woodin.

The principle implies the existence of C-strong cardinals for every class C.

The lower bound for large cardinals implied by Weak Vopěnka's Principle is optimal.

Abstract

We show that Weak Vop\v{e}nka's Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class C there is a C-strong cardinal. Weak Vop\v{e}nka's Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing C-strongness of some cardinal.