In mathematics, a Grothendieck universe is a set U with the following properties: 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.
Grothendieck’s original proposal was to add the following axiom of universes to the usual axioms of set theory: For every set
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.
The concept of a Grothendieck universe can also be defined in a topos.
It is similarly easy to prove that any Grothendieck universe U contains: 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.
Loosely speaking, this is because Grothendieck universes are equivalent to strongly inaccessible cardinals.
More formally, the following two axioms are equivalent: To prove this fact, we introduce the function c(U).
Let u(κ) be the universe of the previous paragraph.
To show that the universe axiom (U) implies the large cardinal axiom (C), choose a cardinal κ. κ is a set, so it is an element of a Grothendieck universe U.
The cardinality of U is strongly inaccessible and strictly larger than that of κ.
This gives another form of the equivalence between Grothendieck universes and strongly inaccessible cardinals: Since the existence of strongly inaccessible cardinals cannot be proved from the axioms of Zermelo–Fraenkel set theory (ZFC), the existence of universes other than the empty set and