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| In [[set theory]], a branch of [[mathematics]], an '''urelement''' or '''ur-element''' (from the [[German language|German]] prefix ''ur-'', 'primordial') is an object (concrete or abstract) that is not a [[Set (mathematics)|set]], but that may be an [[Element (mathematics)|element]] of a set. Urelements are sometimes called "atoms" or "individuals."
| | Adrianne Le is the business name my parents gave me but you can connect with me anything you just like. Vermont has always been my home and I really like every [http://Www.guardian.co.uk/search?q=day+living day living] this site. As a girl what The way we wish like is to have croquet but I would not make it my line of work really. Filing does have been my profession a few time and I'm executing pretty good financially. You can find my website here: http://[http://www.encyclopedia.com/searchresults.aspx?q=prometeu.net prometeu.net]<br><br>My homepage :: [http://prometeu.net triche clash of clans] |
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| == Theory ==
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| There are several different but essentially equivalent ways to treat urelements in a [[first-order theory]].
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| One way is to work in a first-order theory with two sorts, sets and urelements, with ''a'' ∈ ''b'' only defined when ''b'' is a set.
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| In this case, if ''U'' is an urelement, it makes no sense to say
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| :<math>X \in U</math>, | |
| although
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| :<math>U \in X</math>,
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| is perfectly legitimate.
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| This should not be confused with the [[empty set]] where saying
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| :<math>X \in \emptyset</math>
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| is well-formed but false.
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| Another way is to work in a [[Structure (mathematical logic)#Many-sorted structures|one-sorted]] theory with a [[unary relation]] used to distinguish sets and urelements. As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from urelements. Note that in this case, the [[axiom of extensionality]] must be formulated to apply only to objects that are not urelements.
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| This situation is analogous to the treatments of theories of sets and [[Class (set theory)|classes]]. Indeed, urelements are in some sense dual to proper classes: urelements cannot have members whereas proper classes cannot be members. Put differently, urelements are [[Minimal element|minimal]] objects while proper classes are maximal objects by the membership relation (which, of course, is not an order relation, so this analogy is not to be taken literally.)
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| == Urelements in set theory ==
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| The [[Zermelo set theory]] of 1908 included urelements. It was soon realized that in the context of this and closely related [[axiomatic set theory|axiomatic set theories]], the urelements were not needed because they can easily be modeled in a set theory without urelements. Thus standard expositions of the canonical [[axiomatic set theory|axiomatic set theories]] [[Zermelo–Fraenkel set theory|ZF]] and [[ZFC]] do not mention urelements. (For an exception, see Suppes.<ref name="Suppes">{{cite book|last=Suppes|first=Patrick|title=Axiomatic Set Theory|year=1972|publisher=Dover Publ.|location=New York|isbn=0486616304|url=http://store.doverpublications.com/0486616304.html|edition=[Éd. corr. et augm. du texte paru en 1960].|authorlink=Patrick Suppes|accessdate=17 September 2012}}</ref>) [[Axiomatic system#Axiomatizations|Axiomatizations]] of set theory that do invoke urelements include [[Kripke–Platek set theory with urelements]], and the variant of [[Von Neumann–Bernays–Gödel set theory]] described by Mendelson.<ref name="Mendelson">{{cite book|last=Mendelson|first=Elliott|title=Introduction to Mathematical Logic|year=1997|publisher=Chapman & Hall|location=London|isbn=978-0412808302|pages=297–304|url=http://books.google.com/books?id=ZO1p4QGspoYC&lpg=PP1&pg=PT309#v=onepage&q&f=false|edition=4th ed.|accessdate=17 September 2012}}</ref> In [[type theory]], an object of type 0 can be called an urelement; hence the name "atom."
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| Adding urelements to the system [[New Foundations]] (NF) to produce NFU has surprising consequences. In particular, Jensen proved<ref name="Jensen">{{cite journal|last=Jensen|first=Ronald Björn|authorlink=Ronald Jensen|title=On the Consistency of a Slight (?) Modification of Quine's 'New Foundations' |journal=Synthese |date=December 1968 |volume=19 |issue=1/2 |pages=250–264 |url=http://www.jstor.org/stable/20114640 |accessdate=17 September 2012 |publisher=Springer |issn=00397857}}</ref> the [[consistency]] of NFU relative to [[Peano arithmetic]]; meanwhile, the consistency of NF relative to anything remains an open problem. Moreover, NFU remains relatively consistent when augmented with an [[axiom of infinity]] and the [[axiom of choice]]. Meanwhile, the negation of the [[axiom of choice]] is, curiously, an NF theorem. Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF. Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical [[universe]].<ref name="Holmes">Holmes, Randall, 1998. ''[http://math.boisestate.edu/~holmes/holmes/head.pdf Elementary Set Theory with a Universal Set]''. Academia-Bruylant.</ref>
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| == Quine atoms ==
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| An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particular type of set. '''Quine atoms''' are sets that only contain themselves, that is, sets that satisfy the formula ''x'' = {''x''}.<ref name="Forster2003">{{cite book|author=Thomas Forster|title=Logic, Induction and Sets|url=http://books.google.com/books?id=mVeTuaRwWssC&pg=PA199|year=2003|publisher=Cambridge University Press|isbn=978-0-521-53361-4|page=199}}</ref>
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| Quine atoms cannot exist in systems of set theory that include the [[axiom of regularity]], but they can exist in [[non-well-founded set theory]]. ZF set theory with the axiom of regularity removed is compatible with the existence of Quine atoms, although it does not prove that any non-well-founded sets exist. [[Aczel's anti-foundation axiom]] implies there is a unique Quine atom. Other non-well-founded theories may admit many distinct Quine atoms; at the opposite end of the spectrum lies Boffa's [[axiom of superuniversality]], which implies that the distinct Quine atoms form a [[proper class]].<ref name="Aczel1988p57"/>
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| Quine atoms also appear in Quine's [[New Foundations]], which allows more than one such set to exist.<ref>{{citation | first1= Jon |last1= Barwise | first2 = Lawrence S. |last2 = Moss |title= Vicious circles. On the mathematics of non-wellfounded phenomena| series = CSLI Lecture Notes |volume = 60 |publisher = CSLI Publications | year= 1996| isbn= 1575860090 |page=306}}</ref>
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| Quine atoms are the only sets called '''reflexive sets''' by Aczel,<ref name="Aczel1988p57">{{citation
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| |last=Aczel|first= Peter
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| |title=Non-well-founded sets
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| |series= CSLI Lecture Notes|volume= 14|publisher= Stanford University, Center for the Study of Language and Information|year= 1988|page= 57| isbn= 0-937073-22-9
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| |url=http://standish.stanford.edu/pdf/00000056.pdf|mr=0940014}}</ref> although other authors, e.g. [[Jon Barwise]] and Lawrence Moss use the latter term to denote the larger class of sets with the property ''x'' ∈ ''x''.<ref>{{citation | first1= Jon |last1= Barwise | first2 = Lawrence S. |last2 = Moss |title= Vicious circles. On the mathematics of non-wellfounded phenomena| series = CSLI Lecture Notes |volume = 60 |publisher = CSLI Publications | year= 1996| isbn= 1575860090 |page=57}}</ref>
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| == References ==
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| {{reflist}}
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| ==External links==
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| *{{MathWorld|title=Urelement|urlname=Urelement}}
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| [[Category:Urelements| ]]
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Adrianne Le is the business name my parents gave me but you can connect with me anything you just like. Vermont has always been my home and I really like every day living this site. As a girl what The way we wish like is to have croquet but I would not make it my line of work really. Filing does have been my profession a few time and I'm executing pretty good financially. You can find my website here: http://prometeu.net
My homepage :: triche clash of clans