Vopěnka's principle

Definition

Vopěnka's principle asserts that for every proper class of binary relations (each with set-sized domain), there is one elementarily embeddable into another. This cannot be stated as a single sentence of ZFC as it involves a quantification over classes. A cardinal κ is called a Vopěnka cardinal if it is inaccessible and Vopěnka's principle holds in the rank V_κ (allowing arbitrary S ⊂ V_κ as "classes").

Many equivalent formulations are possible. For example, Vopěnka's principle is equivalent to each of the following statements.

For every proper class of simple directed graphs, there are two members of the class with a homomorphism between them.

For any signature Σ and any proper class of Σ-structures, there are two members of the class with an elementary embedding between them.

For every predicate P and proper class S of ordinals, there is a non-trivial elementary embedding j:(V_κ, ∈, P) → (V_λ, ∈, P) for some κ and λ in S.

The category of ordinals cannot be fully embedded in the category of graphs.

Every subfunctor of an accessible functor is accessible.

(In a definable classes setting) For every natural number n, there exists a C⁽ⁿ⁾-extendible cardinal.

Strength

Even when restricted to predicates and proper classes definable in first order set theory, the principle implies existence of Σ_n correct extendible cardinals for every n.

If κ is an almost huge cardinal, then a strong form of Vopenka's principle holds in V_κ:

There is a κ-complete ultrafilter U such that for every {R_i: i < κ} where each R_i is a binary relation and R_i ∈ V_κ, there is S ∈ U and a non-trivial elementary embedding j: R_a → R_b for every a < b in S.