Vopěnka's principle

The Higher Infinite

In mathematics, Vopěnka's principle, named after Petr Vopěnka, is a large cardinal axiom. A cardinal κ is called a Vopěnka cardinal if Vopěnka's principle holds in the rank V_κ (allowing arbitrary S ⊂ V_κ as proper classes).

Vopěnka's principle asserts that for every proper class of binary relations (with set-sized domain), there is one elementarily embeddable into another. Equivalently, for every predicate P and proper class S, there is a non-trivial elementary embedding j:(V_κ, ∈, P) → (V_λ, ∈, P) for some κ and λ in S.

The intuition is that the set-theoretical universe is so large that in every proper class, some members are similar to others, which is formalized through elementary embeddings.

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.

External links

Friedman, Harvey M. (2005), EMBEDDING AXIOMS  gives a number of equivalent definitions of Vopěnka's principle.