Kripke–Platek set theory with urelements

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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

[edit] 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 $Δ 0$ -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.

[edit] Axioms

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

Δ₀-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: $\exists a\, (a=a)$

[edit] Additional assumptions

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

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

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

[edit] Applications

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.

[edit] See also

[edit] 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