A universal set is a set containing every object under consideration in a specified universe of discourse. The complement of every set is then taken relative to the universal set, and the complement of the universal set is the empty set.
In Zermelo-Fraenkel set theory, there is no set of all sets; the collection
of all sets is a proper class.
Indeed, if a set 
contained every set, then the axiom
of separation would give the set
 |
Since 
is universal, 
,
and the definition would imply 
iff 
, which is Russell's
antinomy. Some alternative axiomatic set theories
instead admit a universal set (Holmes 1998).
