If and , then .
If , then .
If , then the power set .
If is a family of elements of , then .
From these axioms, one can deduce the following properties:
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
For every set , there is no infinite descending chain ; 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.
The standard reference for universes is [SGA4].
- 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 http://www.math.mcgill.ca/ archibal/SGA/SGA.htmlavailable on the Web. (It is in French.)