Grothendieck universe

Summary

A Grothendieck universe is a set $U$ that is closed under the usual operations of set theory. These are taking elements, pairs, power sets and unions of families indexed by elements of $U$. Inside a universe one can do ordinary mathematics without leaving $U$. Category theorists use universes to speak of the category of all "small" sets or groups as a set-sized object. Uncountable Grothendieck universes are exactly the sets $V_\kappa$ for inaccessible cardinals $\kappa$, so their existence goes beyond ZFC.