Vopěnka's principle

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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 Vκ (allowing arbitrary SVκ 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:(Vκ, ∈, P) → (Vλ, ∈, 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 Σn correct extendible cardinals for every n.

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

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

External links[edit]

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