A universe $\mathbf{U}$ is a nonempty set satisfying the following axioms:

1. If $x\in \mathbf{U}$ and $y\in x$ then $y\in \mathbf{U}$
2. If $x,y\in \mathbf{U}$ then $\{x,y\}\in \mathbf{U}$
3. If $x\in\mathbf{U}$ then the power set $\mathcal{P}(x)\in\mathbf{U}$
4. If $\{x_i | i\in I\in\mathbf{U}\}$ is a family of elements of $\mathbf{U}$ then $\cup_{i\in I} x_i\in\mathbf{U}$

From these axioms, one can deduce the following properties:

1. If $x\in\mathbf{U}$ then $\{x\}\in\mathbf{U}$
2. If $x$ is a subset of $y\in\mathbf{U}$ then $x\in\mathbf{U}$
3. If $x,y\in\mathbf{U}$ then the ordered pair $(x,y) = \{\{x,y\},x\}$ is in $\mathbf{U}$
4. If $x,y\in\mathbf{U}$ then $x\cup y$ and $x\times y$ are in $\mathbf{U}$
5. If $\{x_i | i\in I\in\mathbf{U}\}$ is a family of elements of $\mathbf{U}$ then the product $\prod_{i\in I} x_i$ is in $\mathbf{U}$
6. If $x\in \mathbf{U}$ then the cardinality of $x$ is strictly less than the cardinality of $\mathbf{U}$ In particular, $\mathbf{U}\notin\mathbf{U}$

In order for uncountable universes to exist, it is necessary to adopt an extra axiom for set theory. This is usually phrased as:

One usually also assumes

Finally, one must be careful when using relations within universes; the details are too technical for Bourbaki to work out (!), but see the appendix to Exposé 1 of [SGA4] for more detail.

The standard reference for universes is [SGA4].

Bibliography

SGA4

Grothendieck et al. Seminaires en Geometrie Algebrique 4, Tome 1, Exposé 1 (or the appendix to Exposé 1, by N. Bourbaki for more detail and a large number of results there described as ``ne pouvant servir à rien''). SGA4 is available on the Web. (It is in French.)