In mathematics, Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarity formalized through elementary embeddings.
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Vopěnka's principle was first introduced by Petr Vopěnka and independently considered by Keisler, and was written up by Solovay, Reinhardt & Kanamori (1978). According to Pudlák (2013, p. 204), Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent. However before publishing his inconsistency proof he found a flaw in it.
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 Vκ (allowing arbitrary S ⊂ Vκ as "classes").
Many equivalent formulations are possible. For example, Vopěnka's principle is equivalent to each of the following statements.
Strength
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 Ri ∈ Vκ, there is S ∈ U and a non-trivial elementary embedding j: Ra → Rb for every a < b in S.