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 universal set leads to a paradox (Russell's paradox) and is consequently not allowed. However, some non-standard variants of set theory include a universal set.
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Notation
There is no standard notation for the universal set of a given set theory. Common symbols include V, U and xi.
Reasons for nonexistence
Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, do not allow for the existence of a universal set. Its existence would cause paradoxes which would make the theory inconsistent.
Russell's paradox
Russell's paradox prevents the existence of a universal set in Zermelo–Fraenkel set theory and other set theories that include Zermelo's axiom of comprehension. This axiom states that, for any formula
that contains exactly those elements x of A that satisfy
Cantor's theorem
A second difficulty with the idea of a universal set concerns the power set of the set of all sets. Because this power set is a set of sets, it would automatically be a subset of the set of all sets, provided that both exist. However, 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.
Theories of universality
The difficulties associated with a universal set can be avoided either by using a variant of set theory in which the axiom of comprehension is restricted in some way, or by using a universal object that is not considered to be a set.
Restricted comprehension
There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V does exist (and
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.
Universal objects that are not sets
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). 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. One difference between a universal set and a universal class is that the universal class does not contain itself, because proper classes cannot be elements of other classes. Russell's paradox does not apply in these theories because the axiom of comprehension operates on sets, not on classes.
The category of sets can also be considered to be a universal object that is, again, not itself a set. It has all sets as elements, and also includes arrows for all functions from one set to another. Again, it does not contain itself, because it is not itself a set.