Vopěnka's principle

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

.

Vopěnka's principle was first introduced by

. 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

and Vopěnka's principle holds in the rank V_κ (allowing arbitrary S ⊂ V_κ as "classes"). [1]

Many equivalent formulations are possible. For example, Vopěnka's principle is equivalent to each of the following statements.

• For every proper class of simple directed graphs, there are two members of the class with a homomorphism between them.^[2]
• For any signature Σ and any proper class of Σ-structures, there are two members of the class with an elementary embedding between them.^[1][2]
• For every predicate P and proper class S of ordinals, there is a non-trivial elementary embedding j:(V_κ, ∈, P) → (V_λ, ∈, P) for some κ and λ in S.^[1]
• The category of ordinals cannot be fully embedded in the category of graphs.^[2]
• Every subfunctor of an accessible functor is accessible.^[2]
• (In a definable classes setting) For every natural number n, there exists a C⁽ⁿ⁾-extendible cardinal.^[3]

Strength

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 Vopěnka'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.

References

• Kanamori, Akihiro (1978), "On Vopěnka's and related principles", Logic Colloquium '77 (Proc. Conf., Wrocław, 1977), Stud. Logic Foundations Math., vol. 96, Amsterdam-New York: North-Holland, pp. 145–153,
  MR 0519809
• Pudlák, Pavel (2013), Logical foundations of mathematics and computational complexity. A gentle introduction, Springer Monographs in Mathematics, Springer,
  MR 3076860
• doi:10.1016/0003-4843(78)90031-1

External links

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

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