Set of all sets

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In set theory as usually formulated, referring to the set of all sets typically leads to a paradox. The reason for this is the form of Zermelo's axiom of separation: for any formula \varphi(x) and set A, the set \{x \in A \mid \varphi(x)\} which contains exactly those elements x of A that satisfy \varphi exists. If the universal set V existed, then we could recover Russell's paradox by considering \{x \in V\mid x\not\in x\}. More generally, for any set A we can prove that \{x \in A\mid x\not\in x\} is not an element of A.

It is natural to want to speak of "all sets" in the usual set theory ZFC, particularly because most versions of this theory do allow us to use quantifiers over all sets (not just quantifiers restricted to particular sets). This is handled by allowing carefully circumscribed mention of V and similar large collections as proper classes. In theories with proper classes the statement V \in V is not true because proper classes cannot be elements.

There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V does exist (and V \in V is true). In these theories, Zermelo's axiom of separation does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way. Examples of such theories are the various versions of New Foundations which are known to be consistent and systems of positive set theory.

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