|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{expert-subject|logic|ex2=philosophy|ex3=mathematics|date=March 2011}}
| | Luke Bryan is a superstar in the making as well as the job expansion very first 2nd to his third studio album, & , will be the evidence. He burst on the scene in 2001 along with his amusing combination of down-property ease of access, film superstar luke bryan 2014 tour tickets ([http://www.ffpjp24.org www.ffpjp24.org]) good seems and words, is defined t inside a key way. The brand new recording Top in the nation graph and #2 about the burst maps, creating it the 2nd [http://okkyunglee.com luke bryan in concert] maximum debut in those days of 1999 for any nation musician. <br><br>The kid of your , understands persistence and determination are important [http://lukebryantickets.citizenswebcasting.com luke bryan s tour] elements when it comes to a prosperous occupation- . His to start with recording, Remain Me, created the best reaches “All My Buddies Say” and “Country Guy,” whilst his energy, Doin’ Issue, found the vocalist-about three directly No. 1 singles: Different Phoning Is often a Excellent Thing.”<br><br>During the slip of 2005, Concerts: Luke [http://lukebryantickets.omarfoundation.org luke bryan tour schedule] Bryan And which in fact had an outstanding set of , including Urban. “It’s almost like you are obtaining a endorsement to look to a higher level, says all those performers that had been a part of the Concert toursabove into a larger level of artists.” It twisted as one of the best organized tours in its ten-season background.<br><br>My site; [http://lukebryantickets.sgs-suparco.org tour luke bryan] |
| The '''Kripke–Platek set theory with urelements''' ('''KPU''') is an [[axiom system]] for [[set theory]] with [[urelement]]s, based on the traditional (urelement-free) [[Kripke-Platek set theory]]. It is considerably weaker than the (relatively) familiar system [[Zermelo–Fraenkel set theory|ZFU]]. The purpose of allowing urelements is to allow large or high-complexity objects (such as [[Baire space (set theory)|the set of all reals]]) to be included in the theory's transitive models without disrupting the usual well-ordering and recursion-theoretic properties of the [[constructible universe]]; KP is so weak that this is hard to do by [[constructible universe#relative constructibility|traditional means]].
| |
| | |
| ==Preliminaries==
| |
| | |
| The usual way of stating the axioms presumes a two sorted first order language <math>L^*</math> with a single binary relation symbol <math>\in</math>.
| |
| Letters of the sort <math>p,q,r,...</math> designate urelements, of which there may be none, whereas letters of the sort <math>a,b,c,...</math> designate sets. The letters <math>x,y,z,...</math> may denote both sets and urelements.
| |
| | |
| The letters for sets may appear on both sides of <math>\in</math>, while those for urelements may only appear on the left, i.e. the following are examples of valid expressions: <math>p\in a</math>, <math>b\in a</math>.
| |
| | |
| The statement of the axioms also requires reference to a certain collection of formulae called <math>\Delta_0</math>-formulae. The collection <math>\Delta_0</math> consists of those formulae that can be built using the constants, <math>\in</math>, <math>\neg</math>, <math>\wedge</math>, <math>\vee</math>, and bounded quantification. That is quantification of the form <math>\forall x \in a</math> or <math> \exists x \in a</math> where <math>a</math> is given set.
| |
| | |
| ==Axioms==
| |
| | |
| The axioms of KPU are the [[universal closure]]s of the following formulae:
| |
| | |
| * [[Axiom of extensionality|Extensionality]]: <math>\forall x (x \in a \leftrightarrow x\in b)\rightarrow a=b</math>
| |
| | |
| * [[Axiom of regularity|Foundation]]: This is an [[axiom schema]] where for every formula <math>\phi(x)</math> we have <math>\exists a \phi(a) \rightarrow \exists a\, (\phi(a) \wedge \forall x\in a\,(\neg \phi(x)))</math>.
| |
| | |
| * [[Axiom of pairing|Pairing]]: <math>\exists a\, (x\in a \land y\in a )</math>
| |
| | |
| * [[Axiom of union|Union]]: <math>\exists a \forall c \in b \forall y\in c\, (y \in a)</math>
| |
| | |
| * [[Axiom schema of predicative separation|Δ<sub>0</sub>-Separation]]: This is again an [[axiom schema]], where for every <math>\Delta_0</math>-formula <math>\phi(x)</math> we have the following <math>\exists a \forall x \,(x\in a \leftrightarrow x\in b \wedge \phi(x) )</math>.
| |
| | |
| * <math>\Delta_0</math>-[[Axiom schema of replacement|Collection]]: This is also an [[axiom schema]], for every <math>\Delta_0</math>-formula <math>\phi(x,y)</math> we have <math>\forall x \in a\exists y\, \phi(x,y)\rightarrow \exists b\forall x \in a\exists y\in b\, \phi(x,y) </math>.
| |
| | |
| * Set Existence: <math>\exists a\, (a=a)</math>
| |
| | |
| ===Additional assumptions===
| |
| Technically these are axioms that describe the partition of objects into sets and urelements.
| |
| | |
| * <math>\forall p \forall a \, (p \neq a)</math>
| |
| | |
| * <math>\forall p \forall x \, (x \notin p)</math>
| |
| | |
| ==Applications==
| |
| | |
| KPU can be applied to the model theory of [[infinitary language]]s. [[model theory|Models]] of KPU considered as sets inside a maximal universe that are [[transitive set|transitive]] as such are called [[admissible set]]s.
| |
| | |
| == See also ==
| |
| * [[Axiomatic set theory]]
| |
| * [[Admissible set]]
| |
| * [[Admissible ordinal]]
| |
| * [[Kripke–Platek set theory]]
| |
| | |
| == References ==
| |
| * {{Citation | last = Barwise | first = Jon | authorlink = Jon Barwise | title = Admissible Sets and Structures | publisher = Springer-Verlag | year = 1975 | isbn = 3-540-07451-1}}.
| |
| * {{Citation | last = Gostanian | first = Richard | title = Constructible Models of Subsystems of ZF | url = http://www.jstor.org/stable/2273185 | journal = [[Journal of Symbolic Logic]] | volume = 45 | pages = 237–250 | year = 1980}}.
| |
| | |
| == External links ==
| |
| <!--* [http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node7.html Admissible Set Theory] (broken?)-->
| |
| * [http://bureau.philo.at/phlo/199703/msg00185.html Logic of Abstract Existence]
| |
| | |
| {{DEFAULTSORT:Kripke-Platek Set Theory With Urelements}}
| |
| [[Category:Systems of set theory]]
| |
| [[Category:Urelements]]
| |
Luke Bryan is a superstar in the making as well as the job expansion very first 2nd to his third studio album, & , will be the evidence. He burst on the scene in 2001 along with his amusing combination of down-property ease of access, film superstar luke bryan 2014 tour tickets (www.ffpjp24.org) good seems and words, is defined t inside a key way. The brand new recording Top in the nation graph and #2 about the burst maps, creating it the 2nd luke bryan in concert maximum debut in those days of 1999 for any nation musician.
The kid of your , understands persistence and determination are important luke bryan s tour elements when it comes to a prosperous occupation- . His to start with recording, Remain Me, created the best reaches “All My Buddies Say” and “Country Guy,” whilst his energy, Doin’ Issue, found the vocalist-about three directly No. 1 singles: Different Phoning Is often a Excellent Thing.”
During the slip of 2005, Concerts: Luke luke bryan tour schedule Bryan And which in fact had an outstanding set of , including Urban. “It’s almost like you are obtaining a endorsement to look to a higher level, says all those performers that had been a part of the Concert toursabove into a larger level of artists.” It twisted as one of the best organized tours in its ten-season background.
My site; tour luke bryan