universe
A universe^{} $\mathrm{\pi \x9d\x90\x94}$ is a nonempty set satisfying the following axioms:

1.
If $x\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$ and $y\beta \x88\x88x$, then $y\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$.

2.
If $x,y\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$, then $\{x,y\}\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$.

3.
If $x\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$, then the power set^{} $\mathrm{\pi \x9d\x92\xab}\beta \x81\u2019(x)\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$.

4.
If $\{{x}_{i}i\beta \x88\x88I\beta \x88\x88\mathrm{\pi \x9d\x90\x94}\}$ is a family of elements of $\mathrm{\pi \x9d\x90\x94}$, then ${\beta \x88\u037a}_{i\beta \x88\x88I}{x}_{i}\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$.
From these axioms, one can deduce the following properties:

1.
If $x\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$, then $\{x\}\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$.

2.
If $x$ is a subset of $y\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$, then $x\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$.

3.
If $x,y\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$, then the ordered pair^{} $(x,y)=\{\{x,y\},x\}$ is in $\mathrm{\pi \x9d\x90\x94}$.

4.
If $x,y\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$, then $x\beta \x88\u037ay$ and $x\Gamma \x97y$ are in $\mathrm{\pi \x9d\x90\x94}$.

5.
If $\{{x}_{i}i\beta \x88\x88I\beta \x88\x88\mathrm{\pi \x9d\x90\x94}\}$ is a family of elements of $\mathrm{\pi \x9d\x90\x94}$, then the product^{} ${\beta \x88\x8f}_{i\beta \x88\x88I}{x}_{i}$ is in $\mathrm{\pi \x9d\x90\x94}$.

6.
If $x\beta \x88\x88\mathrm{\pi \x9d\x90\x94}$, then the cardinality of $x$ is strictly less than the cardinality of $\mathrm{\pi \x9d\x90\x94}$. In particular, $\mathrm{\pi \x9d\x90\x94}\beta \x88\x89\mathrm{\pi \x9d\x90\x94}$.
In order for uncountable universes to exist, it is necessary to adopt an extra axiom for set theory^{}. This is usually phrased as:
Axiom 1.
For every cardinal $\mathrm{\Xi \pm}$, there exists a strongly inaccessible cardinal $\mathrm{\Xi \xb2}\mathrm{>}\mathrm{\Xi \pm}$.
This axiom cannot be proven using the axioms ZFC. But it seems (according to Bourbaki) that it probably cannot be proven not to lead to a contradiction^{}.
One usually also assumes
Axiom 2.
For every set $X$, there is no infinite descending chain $\mathrm{\beta \x8b\u2015}\mathrm{\beta \x88\x88}{x}_{\mathrm{2}}\mathrm{\beta \x88\x88}{x}_{\mathrm{1}}\mathrm{\beta \x88\x88}X$; this is called being artinian.
This axiom does not affect the consistency of ZFC, that is, ZFC is consistent if and only if ZFC with this axiom added is consistent. This is also known as the axiom of foundation, and it is often included with ZFC. If it is not accepted, then one can for all practical purposes restrict oneself to working within the class of artinian sets.
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 ExposΓ© 1 of [SGA4] for more detail.
The standard reference for universes is [SGA4].
References
 SGA4 Grothendieck et al. Seminaires en Geometrie Algebrique 4, Tome 1, ExposΓ© 1 (or the appendix to ExposΓ© 1, by N. Bourbaki for more detail and a large number of results there described as βne pouvant servir Γ rienβ). SGA4 is http://www.math.mcgill.ca/Β archibal/SGA/SGA.htmlavailable on the Web. (It is in French.)
Title  universe 

Canonical name  Universe 
Date of creation  20130322 13:31:13 
Last modified on  20130322 13:31:13 
Owner  archibal (4430) 
Last modified by  archibal (4430) 
Numerical id  8 
Author  archibal (4430) 
Entry type  Definition 
Classification  msc 03E30 
Classification  msc 18A15 
Related topic  Small 