Grothendieck universe
In set theory is a Grothendieck universe (after Alexander Grothendieck ) a lot ( quantities ), in addition, perform the usual set operations on the elements of not, ie it is a model of Zermelo -Fraenkel set theory, the set-theoretic operations ( element relation, power set form ) with those of Zermelo -Fraenkel set theory, in which they are defined, match. The axiom that requires that any amount is an element of a Grothendieck universe, is used in category theory and algebraic geometry and extends the Zermelo -Fraenkel set theory to the Tarski - Grothendieck set theory.
Formal definition
A set is Grothendieck universe, if it satisfies the following axioms:
- : Is element of, then all elements of self and elements of ( transitivity ).
- , With the power set operator called: Is element of, so is the power set of even element of, and thus after the previous condition, all subsets of.
- : Is element of, then the singleton also a member of.
- For each family with and applies :: Associations of elements of elements from again.
- Is not empty.
This definition corresponds to that of P. Gabriel, see ref. Sometimes even the empty set is allowed as a Grothendieck universe, about the SGA.
In other words, is a Grothendieck universe is a model of the form of the two-stage version of ZFC (ie, the replacement axiom scheme is replaced by a single axiom in second order logic with quantification over functions).
Unattainable cardinal numbers
A cardinal number is called (strongly ) inaccessible if the following holds:
- For each set of sets with and
The only known in the Zermelo -Fraenkel set theory ZFC inaccessible cardinal number. The existence of other inaccessible cardinals can not be proved ( the consistency of the same once accepted), but must be postulated by a new axiom in the context of this theory.
The relationship between unattainable cardinal numbers and Grothendieck universes is now manufactured by the following sentence:
For a lot of the following properties are equivalent:
- Is a Grothendieck universe
- There is an unattainable cardinal number such that a and thus apply all of the following equivalent properties: and for each level shall apply:
- (See von Neumann hierarchy)
- (see transitive set )
This is just the cardinality of.
The existence of Grothendieck universes ( except for those with, but containing only finite sets and are therefore not considered to be of interest ) can not be proved within the framework of ZFC set theory in general, but only relatively weak additional assumptions are necessary, namely, the existence of other inaccessible cardinal numbers.
Application in category theory
Assuming the existence of a genuine class of inaccessible cardinals can with the help of Grothendieck universes in category theory statements of any quantity are made.
It is possible at any inaccessible cardinal number assigned to a Grothendieck universe. In order to make a statement of any quantity that a corresponding unattainable cardinal number is required for any amount that is strictly greater than the cardinality of the set, so that a suitable Grothendieck universe exists in which the desired constructions can be carried out.