Vopěnka's principle

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In mathematics, Vopěnka's principle, named after Petr Vopěnka, is a large cardinal axiom.

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. A cardinal κ is Vopěnka if and only if Vopěnka's principle holds in V_κ (allowing arbitrary S ⊂ V_κ as proper classes).

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.

A number of equivalent definitions of Vopěnka's principle can be found in
http://www.cs.nyu.edu/pipermail/fom/2005-August/009023.html

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.