Grothendieck Universe

In mathematics, a Grothendieck universe is a set U with the following properties:

  1. If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.)
  2. If x and y are both elements of U, then {x,y} is an element of U.
  3. If x is an element of U, then P(x), the power set of x, is also an element of U.
  4. If is a family of elements of U, and if I is an element of U, then the union is an element of U.

Elements of a Grothendieck universe are sometimes called small sets.

A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, uncountable Grothendieck universes provide models of set theory with the natural ∈-relation, natural powerset operation etc.) As an example, we will prove an easy proposition.

The axiom of Grothendieck universes says that every set is an element of a Grothendieck universe.

Proposition. If and, then .
Proof. because . because, so .

It is similarly easy to prove that any Grothendieck universe U contains:

  • All singletons of each of its elements,
  • All products of all families of elements of U indexed by an element of U,
  • All disjoint unions of all families of elements of U indexed by an element of U,
  • All intersections of all families of elements of U indexed by an element of U,
  • All functions between any two elements of U, and
  • All subsets of U whose cardinal is an element of U.

In particular, it follows from the last axiom that if U is non-empty, it must contain all of its finite subsets and a subset of each finite cardinality. One can also prove immediately from the definitions that the intersection of any class of universes is a universe.

The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry.

Read more about Grothendieck Universe:  Grothendieck Universes and Inaccessible Cardinals

Other articles related to "grothendieck universes, grothendieck universe, universe, universes":

Category Of Sets - Foundations For The Category of Sets
... Another solution is to assume the existence of Grothendieck universes ... Roughly speaking, a Grothendieck universe is a set which is itself a model of ZF(C) (for instance if a set belongs to a universe, its elements and its powerset will belong to the universe) ... The existence of Grothendieck universes (other than the empty set and the set of all hereditarily finite sets) is not implied by the usual ZF axioms it is an additional ...
Grothendieck Universes and Inaccessible Cardinals
... There are two simple examples of Grothendieck universes The empty set, and The set of all hereditarily finite sets ... Loosely speaking, this is because Grothendieck universes are equivalent to strongly inaccessible cardinals ... (U) For each set x, there exists a Grothendieck universe U such that x ∈ U ...
Universe (mathematics) - In Category Theory
... There is another approach to universes which is historically connected with category theory ... This is the idea of a Grothendieck universe ... Roughly speaking, a Grothendieck universe is a set inside which all the usual operations of set theory can be performed ...

Famous quotes containing the word universe:

    The universe is made of stories,
    not of atoms.
    Muriel Rukeyser (1913–1980)