Editing Universal set (section)

==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 {{mvar|V}} does exist (and <math>V \in V</math> 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,<ref>{{harvtxt|Church|1974|p=308}}. See also {{harvtxt|Forster|1995|p=136}}, {{harvtxt|Forster|2001|p=17}}, and {{harvtxt|Sheridan|2016}}.</ref> but this is not possible for Oberschelp's, since in it the singleton function is provably a set,{{sfnp|Oberschelp|1973|p=40}} which leads immediately to paradox in New Foundations.{{sfnp|Holmes|1998|p=110}}

Another example is [[positive set theory]], where the axiom of comprehension is restricted to hold only for the [[positive formula]]s (formulas that do not contain negations). Such set theories are motivated by notions of closure in topology.

===Universal objects that are not sets===
{{main|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 {{mvar|V}} and similar large collections as [[Class (set theory)|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.

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.