Totally disconnected space - Revision history

en>Paul August: Reverted edits by 103.27.8.43 (talk) to last version by Trappist the monk

2014-11-11T19:51:53Z

Reverted edits by 103.27.8.43 (talk) to last version by Trappist the monk

← Older revision		Revision as of 21:51, 11 November 2014
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en>Wavelength: inserting 1 hyphen: —> "Zero-dimensional"—wikt:zero-dimensional

2014-02-21T16:17:01Z

inserting 1 hyphen: —> "Zero-dimensional"—wikt:zero-dimensional

Show changes

en>Danneks: /* Constructing a disconnected space */

2013-10-16T13:29:32Z

Constructing a disconnected space

← Older revision		Revision as of 15:29, 16 October 2013
Line 1:		Line 1:
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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 {{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. 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. 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]]

en>Jowa fan: /* References */ give MR link to the original publication; the MR page for the Dover edition isn't very informative

2012-04-20T05:12:13Z

References: give MR link to the original publication; the MR page for the Dover edition isn't very informative

New page

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