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

- 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.) - If
*x*and*y*are both elements of*U*, then {*x*,*y*} is an element of*U*. - If
*x*is an element of*U*, then*P(x)*, the power set of*x*, is also an element of*U*. - 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

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