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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]]

Revision as of 18:17, 21 February 2014

In true of a locksmith service, this type of appeal could be general. Finally, connect it to a stereo or set of powered speakers and you're simply geared up to go. "2ND Bose Tabletop System Gets Wireless PC Streaming". As we walk, music walks around, adding rhythm to the steps; and as we lower the road, the linear progression of the song coincides with the passing scenery. 0 full speed so as to make the ease transfer of music. While the songs in Peru and America appears from the existence from the first inhabitants of the region, this is devoted to the music 'West' of Peru, which began with all the European invasion.

One measure that's increasing in popularity employs music. It is not any doubt that Perry was happy with her appearance. This early strategy for rapping was soon enhanced to add manual manipulation of the sound system to heighten the dance experience. The audiences who liked the products earlier started taking it in a negative way as well as the experts began to give negative comments. The Sound - Dock shares the Active EQ found for the Quiet - Comfort group of headphones, the EM shielding found on Bose's computer speaker family, the compression circuitry and also the integrated signal processing on existing Bose speaker systems. Certainly you do not need to be the 'best' at all of such things (you ought to find other people who are strong in the areas you are weak in) and yes it isn't always good to try and do everything on the own.



The lyrics and flow of bluegrass music are incredibly similar to that of country music. Even when someone does, it is possible to litigate him in a very lawsuit. Polynesian music is a fun and vibrant musical style originating from the Polynesian islands with the South Pacific. Well, this post is obviously for you as I have a tendency to share my experiences with a very reliable and affordable music studio software and just how I was capable of find the correct software for my music studio. But that is just not the problem; principle thing will be the destruction of one.

As mentioned in a previous article I wrote, Music Video. For example, one can assume that mathematics may also develop music. Last although not least, Stream - Furious could be the great music application available for the HTC Sensation. For the songs box and antique collector, there are many types and styles to decide on from. If your teacher have the identical concept and notion related to music, then you'll see the classes would be considerably more enjoyable and you will be able to learn quite a lot apart from your regular lessons.

There certainly are a variety of legal download sites about the Internet. Two similar Christian vocal groups are Take 6 and Acappella. When she began lessons,she ran round the practice room and hid. Anyone can learn them and use the crooks to advance their music careers (for example you). My teaching philosophy is always that guitar, piano or voice lessons might help children in achieving goals, increasing their self confidence, and developing technical competence. The positive is connected on the positive terminal along with the negative is linked to negative.



Tomatis discovered how classical music, especially from your famous composer Mozart, could significantly relieve the "ringing in the ears" that is called tinnitus. Next, once they get within viewing distance of the angel music boxes, their senses get into toxic upset and they generally explode. Join professional associations like The National Association of Record Industry Professionals (NARIP). Music lovers can get this mobile without second thoughts. These .U_O-4fmSzzI titles might be downloaded for guitars, piano, vocal, violin and also other instruments. The different language can have different karaokeâ&#128;&#153;s like Panjabi karaoke, Bengali karaoke, Hindi karaoke etc.

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