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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." | |||
== Theory == | |||
There are several different but essentially equivalent ways to treat urelements in a [[first-order theory]]. | |||
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. | |||
In this case, if ''U'' is an urelement, it makes no sense to say | |||
:<math>X \in U</math>, | |||
although | |||
:<math>U \in X</math>, | |||
is perfectly legitimate. | |||
This should not be confused with the [[empty set]] where saying | |||
:<math>X \in \emptyset</math> | |||
is well-formed but false. | |||
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. | |||
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.) | |||
== Urelements in set theory == | |||
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." | |||
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> | |||
== Quine atoms == | |||
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> | |||
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"/> | |||
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> | |||
Quine atoms are the only sets called '''reflexive sets''' by Aczel,<ref name="Aczel1988p57">{{citation | |||
|last=Aczel|first= Peter | |||
|title=Non-well-founded sets | |||
|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 | |||
|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> | |||
== References == | |||
{{reflist}} | |||
==External links== | |||
*{{MathWorld|title=Urelement|urlname=Urelement}} | |||
[[Category:Urelements| ]] | |||
Revision as of 17:32, 2 February 2014
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object (concrete or abstract) that is not a set, but that may be an element of a set. Urelements are sometimes called "atoms" or "individuals."
Theory
There are several different but essentially equivalent ways to treat urelements in a first-order theory.
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. In this case, if U is an urelement, it makes no sense to say
although
is perfectly legitimate.
This should not be confused with the empty set where saying
is well-formed but false.
Another way is to work in a 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.
This situation is analogous to the treatments of theories of sets and 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 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.)
Urelements in set theory
The Zermelo set theory of 1908 included urelements. It was soon realized that in the context of this and closely related 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 theories ZF and ZFC do not mention urelements. (For an exception, see Suppes.[1]) 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.[2] In type theory, an object of type 0 can be called an urelement; hence the name "atom."
Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences. In particular, Jensen proved[3] 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.[4]
Quine atoms
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}.[5]
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.[6]
Quine atoms also appear in Quine's New Foundations, which allows more than one such set to exist.[7]
Quine atoms are the only sets called reflexive sets by Aczel,[6] 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.[8]
References
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External links
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010