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Concept
Grothendieck universe
Summary
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 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.
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.). Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry.
The existence of a nontrivial Grothendieck universe goes beyond the usual axioms of Zermelo–Fraenkel set theory; in particular it would imply the existence of strongly inaccessible cardinals.
Tarski–Grothendieck set theory is an axiomatic treatment of set theory, used in some automatic proof systems, in which every set belongs to a Grothendieck universe.
The concept of a Grothendieck universe can also be defined in a topos.
As an example, we will prove an easy proposition.
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.
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