Definition:Grothendieck Universe

Definition
A Grothendieck universe is a universe $\mathbb U$ of sets in the sense of the Zermelo-Fraenkel axioms with the following properties:


 * 1. $\mathbb U$ is a transitive set: if $u \in \mathbb U$ and $x \in u$ then $x \in \mathbb U$


 * 2. If $u,v \in \mathbb U$ then $\{u,v\} \in \mathbb U$


 * 3. If $u \in \mathbb U$ then the power set $\mathcal P \left({u}\right) \in \mathbb U$


 * 4. If $A \in \mathbb U$, and $\{u_\alpha : \alpha \in A\}$ is a collection of elements of $\mathbb U$, then $\displaystyle \bigcup_{\alpha \in A}u_\alpha \in \mathbb U$