Universal Set

In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, the conception of a set of all sets leads to a paradox. The reason for this lies with Zermelo's axiom of comprehension: for any formula and set A, there exists a set

which contains exactly those elements x of A that satisfy exists. If the universal set V existed and the axiom of separation applied to it, then Russell's paradox would arise from

.

More generally, for any set A we can prove that

is not an element of A.

A second difficulty is that the power set of the set of all sets would be a subset of the set of all sets, provided that both exist. This conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality than the set itself.

The idea of a universal set seems intuitively desirable in the Zermelo–Fraenkel set theory, particularly because most versions of this theory do allow the use of quantifiers over all sets (see universal quantifier). This is handled by allowing carefully circumscribed mention of V and similar large collections as proper classes. In theories in which the universe is a proper class, is not true because proper classes cannot be elements.