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.
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is for