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 ${\displaystyle \{x,y\}}$ 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 ${\displaystyle \{x_{\alpha }\}_{\alpha \in I}}$ is a family of elements of U, and if I is an element of U, then the union ${\textstyle \bigcup _{\alpha \in I}x_{\alpha }}$ 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. Grothendieck’s original proposal was to add the following axiom of universes to the usual axioms of set theory: For every set ${\displaystyle s}$, there exists a universe ${\displaystyle U}$ that contains ${\displaystyle s}$, i.e., ${\displaystyle s\in U}$.

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.^[1]