Vopenka's principle
Encyclopedia
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 S ⊂ Vκ 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 cardinal
s for every n.
If κ is an almost huge cardinal, then a strong form of Vopenka's principle holds in Vκ:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Vopěnka's principle, named after Petr Vopěnka
Petr Vopenka
Petr Vopěnka is a Czech mathematician. In the early seventies, he established the Alternative Set Theory , which he subsequently developed in a series of articles and monographs...
, is a large cardinal axiom. A cardinal κ is called a Vopěnka cardinal if Vopěnka's principle holds in the rank Vκ (allowing arbitrary S ⊂ Vκ 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 cardinal
Extendible cardinal
In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. A cardinal number κ is called η-extendible if for some λ there is a nontrivial elementary embedding j ofinto...
s 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 ultrafilterUltrafilterIn the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
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.