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.