Vopěnka's principle

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".