- Kripke–Platek set theory with urelements
The Kripke–Platek set theory with urelements (KPU) is an
axiom systemfor set theorywith urelements that is considerably weaker than the familiar system ZF.
The usual way of stating the axioms presumes a two sorted first order language with a single binary relation symbol . Letters of the sort designate urelements, of which there may be none, whereas letters of the sort designate sets. The letters may denote both sets and urelements.
The letters for sets may appear on both sides of , while those for urelements may only appear on the left, i.e. the following are examples of valid expressions: , .
The statement of the axioms also requires reference to a certain collection of formulae called -formulae. The collection consists of those formulae that can be built using the constants, , , , , and bounded quantification. That is quantification of the form or where is given set.
The axioms of KPU are the
universal closures of the following formulae:
* Foundation: This is an
axiom schemawhere for every formula we have .
* Δ0-Separation: This is again an
axiom schema, where for every -formula we have the following .
* -Collection: This is also an
axiom schema, for every -formula we have .
* Set Existence:
Technically these are axioms that describe the partition of objects into sets and urelements.
Axiomatic set theory
Kripke–Platek set theory
* 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
* [http://bureau.philo.at/phlo/199703/msg00185.html Logic of Abstract Existence]
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