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			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]]

en>Arkuse at 20:29, 15 August 2012

2012-08-15T20:29:03Z

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130.226.137.71: /* Construction */

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en>Arkuse at 20:29, 15 August 2012