Definition 1.1.3 (Grothendieck Universe).label A Grothendieck universe is a pure set $U$ such that:

1. (GU1)

   $\fall u\in U,t\in u,t\in U$

2. (GU2)

   $\fall u\in U,\cali{P}(u)\in U$

3. (GU3)

   $\emp\in U$

4. (GU4)

   $\fall I\in U$ and functions $u:I\to U,\cups{}{i\in I }u_{i}\in U$

An element of $U$ is called a $U$-small set, while a subset of $U$ is called $U$-moderate. Every $U$-small set is $U$-moderate by (GU1).