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Reference
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 Russell's paradox and is consequently not allowed. However, some non-standard variants of set theory include a universal set.
Notation
There is no standard notation for the universal set of a given set theory. Common symbols include V, U and ?.[citation needed]
Reasons for nonexistence Many set theories do not allow for the existence of a universal set. For example, it is directly contradicted by the axioms such as the axiom of regularity and its existence would imply inconsistencies. The standard Zermelo–Fraenkel set theory is instead based on the cumulative hierarchy.
Russell's paradox
Main article: Russell's paradox
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 {\displaystyle V\in V}V\in V is true). 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. A set theory containing a universal set is necessarily a non-well-founded set theory. 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,but this is not possible for Oberschelp's, since in it the singleton function is provably a set,[4] which leads immediately to paradox in New Foundations.
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
Main article: Universe (mathematics)
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.[citation needed] 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.
https://en.wikipedia.org/wiki/Universal_set
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