Kripke–Platek set theory with urelements

The Kripke–Platek set theory with urelements (KPU) is an axiom system for set theory with urelements, based on the traditional (urelement-free) Kripke–Platek set theory. 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; 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: $$\forall x (x \in a \leftrightarrow x\in b)\rightarrow a=b$$
 * Foundation: This is an axiom schema 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: $$\exists a\, (x\in a \land y\in a )$$
 * Union: $$\exists a \forall c \in b. \forall y\in c (y \in a)$$
 * &Delta;0-Separation: This is again an axiom schema, where for every $$\Delta_0$$-formula $$\phi(x)$$ we have the following $$\exists a \forall x \,(x\in a \leftrightarrow x\in b \wedge \phi(x) )$$.
 * &Delta;0-SCollection: This is also an axiom schema, 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 of KPU considered as sets inside a maximal universe that are transitive as such are called admissible sets.