Kripke–Platek set theory with urelements

From Wikipedia, the free encyclopedia

The Kripke–Platek set theory with urelements (KPU) is an axiom system for set theory with urelements that is considerably weaker than the familiar system ZF.

1 Preliminaries
2 Axioms
- 2.1 Additional assumptions
3 Applications
4 See also
5 References
6 External links

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 $Δ 0$ -formulae. The collection $Δ 0$ 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.

The axioms of KPU are the universal closures of the following formulae:

Extensionality: $\forall x (x \in a \leftrightarrow x\in b)\rightarrow a=b$

Foundation: This is an axiom schema where for every formula $φ(x)$ we have $\exists x \phi(x) \rightarrow \exists x\, (\phi(x) \wedge \forall y\in x\,(\neg \phi(y)))$ .

Pairing: $\exists a\, (x\in a \land y\in a )$

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

$Δ 0$ -Separation: This is again an axiom schema, where for every $Δ 0$ -formula $φ(x)$ we have the following $\exists a \forall x \,(x\in a \leftrightarrow x\in b \wedge \phi(x) )$ .

$Δ 0$ -Collection: This is also an axiom schema, for every $Δ 0$ -formula $φ(x, y)$ 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: $\exists a\, (a=a)$

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

$\forall p \forall a \, (p \neq a)$

$\forall p \forall x \, (x \notin p)$

KPU can be applied to the model theory of infinitary languages. Models of KPU considered as sets inside a maximal universe that are transitive as such are called admissible sets.

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

Logic of Abstract Existence

Retrieved from "http://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory_with_urelements"

Categories: Systems of set theory | Urelements

Searching reference...

Reference

Sources

Kripke–Platek set theory with urelements

From Wikipedia, the free encyclopedia

Contents

Top Matching Results

Sponsored Links
This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Searching reference...

Reference

Sources

Kripke–Platek set theory with urelements

From Wikipedia, the free encyclopedia

Contents

Top Matching Results

Sponsored Links This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Sponsored Links
This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.