mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Vopěnka's principle is a

large cardinal In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...

axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every

proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...

, some members are similar to others, with this similarity formalized through

elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one oft ...

s. Vopěnka's principle was first introduced by

Petr Vopěnka Petr Vopěnka (16 May 1935 – 20 March 2015) was a Czech people, Czech mathematician. In the early seventies, he developed alternative set theory (i.e. alternative to the classical Cantor theory), which he subsequently developed in a series of ...

and independently considered by

H. Jerome Keisler Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski a ...

, and was written up by . According to , 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

binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...

s (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 Inaccessible Island is a volcanic island located in the South Atlantic Ocean, south-west of Tristan da Cunha. Its highest point, Swale's Fell, reaches , and the island is in area. The volcano was last active approximately one million years a ...

and Vopěnka's principle holds in the rank ''V''_κ (allowing arbitrary ''S'' ⊂ ''V''_κ as "classes"). 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. * For any

signature A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...

''Σ'' and any proper class of ''Σ''-structures, there are two members of the class with an elementary embedding between them. * 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''. * The

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

of ordinals cannot be fully embedded in the category of graphs. * Every subfunctor of an accessible functor is accessible. * (In a definable classes setting) For every natural number ''n'', there exists a ''C⁽ⁿ⁾''-extendible cardinal.

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 In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...

''U'' such that for every 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

* * *

External links

* gives a number of equivalent definitions of Vopěnka's principle. Large cardinals Mathematical principles {{settheory-stub