- Kripke–Platek set theory with urelements
The

**Kripke–Platek set theory with urelements**(**KPU**) is anaxiom system forset theory withurelement s that is considerably weaker than the familiar system ZF.**Preliminaries**The usual way of stating the axioms presumes a two sorted first order language $L^*$ with a single binary relation symbol $in$. Letters of the sort $p,q,r,...$ designate urelements, of which there may be none, whereas letters of the sort $a,b,c,...$ designate sets. The letters $x,y,z,...$ may denote both sets and urelements.

The letters for sets may appear on both sides of $in$, while those for urelements may only appear on the left, i.e. the following are examples of valid expressions: $pin\; a$, $bin\; a$.

The statement of the axioms also requires reference to a certain collection of formulae called $Delta\_0$-formulae. The collection $Delta\_0$ consists of those formulae that can be built using the constants, $in$, $eg$, $wedge$, $vee$, and bounded quantification. That is quantification of the form $forall\; x\; in\; a$ or $exists\; x\; in\; a$ where $a$ is given set.

**Axioms**The axioms of KPU are the

universal closure s of the following formulae:* Extensionality: $forall\; x\; (x\; in\; a\; leftrightarrow\; xin\; b)\; ightarrow\; a=b$

* Foundation: This is an

axiom schema where for every formula $phi(x)$ we have $exists\; x\; phi(x)\; ightarrow\; exists\; x,\; (phi(x)\; wedge\; forall\; yin\; x,(\; eg\; phi(y)))$.* Pairing: $exists\; a,\; (xin\; a\; land\; yin\; a\; )$

* Union: $exists\; a\; forall\; x\; in\; b\; forall\; yin\; x,\; (y\; in\; a)$

* Δ

_{0}-Separation: This is again anaxiom schema , where for every $Delta\_0$-formula $phi(x)$ we have the following $exists\; a\; forall\; x\; ,(xin\; a\; leftrightarrow\; xin\; b\; wedge\; phi(x)\; )$.* $Delta\_0$-Collection: This is also an

axiom schema , for every $Delta\_0$-formula $phi(x,y)$ we have $forall\; x\; in\; aexists\; y,\; phi(x,y)\; ightarrow\; exists\; bforall\; x\; in\; aexists\; yin\; b,\; phi(x,y)$.* Set Existence: $exists\; a,\; (a=a)$

**Additional assumptions**Technically these are axioms that describe the partition of objects into sets and urelements.

* $forall\; p\; forall\; a\; ,\; (p\; eq\; a)$

* $forall\; p\; forall\; x\; ,\; (x\; otin\; p)$

**Applications**KPU can be applied to the model theory of

infinitary language s. Models of KPU considered as sets inside a maximal universe that are transitive as such are calledadmissible set s.**See also***

Axiomatic set theory

*Admissible ordinal

*Kripke–Platek set theory **References*** Gostanian, Richard, 1980, "Constructible Models of Subsystems of ZF," "Journal of Symbolic Logic 45" (2): .

*Jon Barwise , "Admissible Sets and Structures". Springer Verlag. ISBN 3540074511**External links*** [

*http://bureau.philo.at/phlo/199703/msg00185.html Logic of Abstract Existence*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Kripke–Platek set theory**— The Kripke–Platek axioms of set theory (KP) (IPAEng|ˈkrɪpki ˈplɑːtɛk) are a system of axioms of axiomatic set theory, developed by Saul Kripke and Richard Platek. The axiom system is written in first order logic; it has an infinite number of… … Wikipedia**Set theory**— This article is about the branch of mathematics. For musical set theory, see Set theory (music). A Venn diagram illustrating the intersection of two sets. Set theory is the branch of mathematics that studies sets, which are collections of objects … Wikipedia**List of set theory topics**— Logic portal Set theory portal … Wikipedia**Urelement**— In set theory, a branch of mathematics, an urelement or ur element (from the German prefix ur , primordial ) is an object (concrete or abstract) which is not a set, but that may be an element of a set. Urelements are sometimes called atoms or… … Wikipedia**List of mathematics articles (K)**— NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) … Wikipedia**List of mathematical logic topics**— Clicking on related changes shows a list of most recent edits of articles to which this page links. This page links to itself in order that recent changes to this page will also be included in related changes. This is a list of mathematical logic … Wikipedia**Axiom schema of specification**— For the separation axioms in topology, see separation axiom. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or… … Wikipedia**KPU**— is an abbreviation that can mean:* Kripke–Platek set theory with urelements, an axiom system for set theory * Kwantlen Polytechnic University, University in British Columbia. * [http://www.kpu.ac.kr/mainEng/index.do Korea Polytechnic University] … Wikipedia**Mathematical logic**— (also known as symbolic logic) is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic.[1] The field includes both the mathematical study of logic and the… … Wikipedia