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In [[set theory]], a '''universal set''' is a [[Set (mathematics)|set]] which contains all objects, including itself.<ref>Forster 1995 p. 1.</ref> In [[set theory]] as usually formulated, the conception of a set of all sets leads to a [[paradox]]. However, some non-standard variants of set theory include a universal set.
 
==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 <math>\varphi(x)</math> and any set {{mvar|A}}, there exists another set
:<math>\{x \in A \mid \varphi(x)\}</math>
that contains exactly those elements {{mvar|x}} of {{mvar|A}} that satisfy <math>\varphi</math>.  If a universal set&nbsp;{{mvar|V}} existed and the axiom of comprehension could be applied to it, then
there would also exist another set <math>\{x \in V\mid x\not\in x\}</math>, the set of all sets that do not contain themselves. However, as [[Bertrand Russell]] observed, this set is paradoxical. If it contains itself, then it should not contain itself, and vice versa. For this reason, it cannot exist.
 
===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 {{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 [[:de:Arnold Oberschelp|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>Church 1974 p. 308. See also Forster 1995 p. 136 or 2001 p. 17.</ref> but this is not possible for Oberschelp’s, since in it the singleton function is provably a set,<ref>Oberschelp 1973 p. 40.</ref> which leads immediately to paradox in New Foundations.<ref>Holmes 1998 p. 110.</ref>
 
===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. One difference between a universal set and a universal class is that the universal class does not contain itself, because [[proper class]]es 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.
 
== Notes ==
{{Reflist}}
 
== References ==
* [[Alonzo Church]] (1974). [http://books.google.com/books?id=6GFNxtPAK8UC&hl=en&output=reader&pg=GBS.PA297 “Set Theory with a Universal Set,”] [http://www.ams.org/books/pspum/025/pspum025-endmatter.pdf ''Proceedings of the Tarski Symposium. Proceedings of Symposia in Pure Mathematics XXV,''] ed. L. Henkin, American Mathematical Society, pp.&nbsp;297–308.
* {{cite book | author=[[T. E. Forster]] | title= Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford Logic Guides 31) | publisher=Oxford University Press | year=1995 | isbn=0-19-851477-8}}
* [http://www.dpmms.cam.ac.uk/~tf/ T. E. Forster] (2001). [http://www.dpmms.cam.ac.uk/~tf/church2001.ps “Church’s Set Theory with a Universal Set.”]
* [http://math.boisestate.edu/~holmes/holmes/setbiblio.html Bibliography: Set Theory with a Universal Set], originated by T. E. Forster and maintained by Randall Holmes at Boise State University.
* [http://math.boisestate.edu/~holmes Randall Holmes] (1998). ''[http://math.boisestate.edu/~holmes/holmes/head.ps Elementary Set theory with a Universal Set,]'' volume 10 of the Cahiers du Centre de Logique, Academia, Louvain-la-Neuve (Belgium).
* [[:de:Arnold Oberschelp|Arnold Oberschelp]] (1973). “Set Theory over Classes,” ''Dissertationes Mathematicae'' 106.
* [[Willard Van Orman Quine]] (1937) “New Foundations for Mathematical Logic,” ''American Mathematical Monthly'' 44, pp.&nbsp;70–80.
 
== External links ==
*{{MathWorld |title=Universal Set |id=UniversalSet }}
 
{{Set theory}}
 
{{DEFAULTSORT:Universal Set}}
[[Category:Basic concepts in set theory]]
[[Category:Set families]]
[[Category:Paradoxes of naive set theory]]
[[Category:Systems of set theory]]
[[Category:Wellfoundedness]]
[[Category:Self-reference]]

Latest revision as of 20:51, 11 November 2014

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