Grothendieck universes are equivalent to strongly inaccessible cardinals.
More formally, the following two axioms are equivalent:
(U) For all sets x, there exists a Grothendieck universe U such that x ∈ U.
(C) For all cardinals κ, there is a strongly inaccessible cardinal λ which is strictly larger than κ.

Since the existence of strongly inaccessible cardinals is logically independent of ZFC
(the axioms of Zermelo-Fraenkel set theory plus the axiom of choice),
the existence of universes is also logically independent.
The idea of universes is due to Alexander Grothendieck, who used them as a way of
avoiding proper classes in algebraic geometry.
http://en.wikipedia.org/wiki/Grothendieck_universe