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The Kripke–Platek set theory with urelements (KPU) is an axiom system for
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
with urelements, based on the traditional (urelement-free)
Kripke–Platek set theory The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of Zermelo–Fraenkel set theory (ZFC) and is considerably weak ...
. It is considerably weaker than the (relatively) familiar system ZFU. The purpose of allowing urelements is to allow large or high-complexity objects (such as 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 In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular Class (set theory), class of Set (mathematics), sets that can be described entirely in terms of simpler sets. L is the un ...
; KP is so weak that this is hard to do by traditional means.


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: p\in a, b\in 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, \neg, \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 closures of the following formulae: *
Extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equality (mathematics), equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned wi ...
: \forall x (x \in a \leftrightarrow x\in b)\rightarrow a=b * Foundation: This is an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variabl ...
where for every formula \phi(x) we have \exists a. \phi(a) \rightarrow \exists a (\phi(a) \wedge \forall x\in a\,(\neg \phi(x))). *
Pairing In mathematics, a pairing is an ''R''- bilinear map from the Cartesian product of two ''R''- modules, where the underlying ring ''R'' is commutative. Definition Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be '' ...
: \exists a\, (x\in a \land y\in a ) * Union: \exists a \forall c \in b. \forall y\in c (y \in a) * Δ0-Separation: This is again an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variabl ...
, where for every \Delta_0-formula \phi(x) we have the following \exists a \forall x \,(x\in a \leftrightarrow x\in b \wedge \phi(x) ). * Δ0-SCollection: This is also an
axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variabl ...
, for every \Delta_0-formula \phi(x,y) we have \forall x \in a.\exists y. \phi(x,y)\rightarrow \exists b\forall x \in a.\exists y\in 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 \neq a) * \forall p \forall x (x \notin p)


Applications

KPU can be applied to the model theory of infinitary languages.
Models A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided int ...
of KPU considered as sets inside a maximal universe that are transitive as such are called admissible sets.


See also

*
Axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
* Admissible set * Admissible ordinal *
Kripke–Platek set theory The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of Zermelo–Fraenkel set theory (ZFC) and is considerably weak ...


References

* . * .


External links

* * {{DEFAULTSORT:Kripke-Platek Set Theory With Urelements Systems of set theory Urelements