[1] In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist.
There are several different arguments for its non-existence, based on different choices of axioms for set theory.
[2] This paradox prevents the existence of a universal set in set theories that include either Zermelo's axiom of restricted comprehension, or the axiom of regularity and axiom of pairing.
(constructed using pairing) necessarily contains an element disjoint from
Because a universal set would necessarily contain itself, it cannot exist under these axioms.
[3] Russell's paradox prevents the existence of a universal set in set theories that include Zermelo's axiom of restricted comprehension.
, it would state the existence of Russell's paradoxical set, giving a contradiction.
It was this contradiction that led the axiom of comprehension to be stated in its restricted form, where it asserts the existence of a subset of a given set rather than the existence of a set of all sets that satisfy a given formula.
[2] When the axiom of restricted comprehension is applied to an arbitrary set
, because if it were it would be included as a member of itself, by its definition, contradicting the fact that it cannot contain itself.
This indeed holds even with predicative comprehension and over intuitionistic logic.
In these theories, Zermelo's axiom of comprehension does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way.
The most widely studied set theory with a universal set is Willard Van Orman Quine's New Foundations.
Alonzo Church and Arnold Oberschelp also published work on such set theories.
Church speculated that his theory might be extended in a manner consistent with Quine's,[4] but this is not possible for Oberschelp's, since in it the singleton function is provably a set,[5] which leads immediately to paradox in New Foundations.
[6] Another example is positive set theory, where the axiom of comprehension is restricted to hold only for the positive formulas (formulas that do not contain negations).
Such set theories are motivated by notions of closure in topology.
One way of allowing an object that behaves similarly to a universal set, without creating paradoxes, is to describe V and similar large collections as proper classes rather than as sets.
Russell's paradox does not apply in these theories because the axiom of comprehension operates on sets, not on classes.