Grothendieck universe
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.
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.
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. One can also prove immediately from the definitions that the intersection of any class of universes is a universe.
There are two simple examples of Grothendieck universes:
Other examples are more difficult to construct. 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). Define:
where by |x| we mean the cardinality of x. Then for any universe U, c(U) is either zero or strongly inaccessible. Assuming it is non-zero, it is a strong limit cardinal because the power set of any element of U is an element of U and every element of U is a subset of U. To see that it is regular, suppose that cλ is a collection of cardinals indexed by I, where the cardinality of I and of each cλ is less than c(U). Then, by the definition of c(U), I and each cλ can be replaced by an element of U. The union of elements of U indexed by an element of U is an element of U, so the sum of the cλ has the cardinality of an element of U, hence is less than c(U). By invoking the axiom of foundation, that no set is contained in itself, it can be shown that c(U) equals |U|; when the axiom of foundation is not assumed, there are counterexamples (we may take for example U to be the set of all finite sets of finite sets etc. of the sets xα where the index α is any real number, and xα = {xα} for each α. Then U has the cardinality of the continuum, but all of its members have finite cardinality and so ; see Bourbaki's article for more details).
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