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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, 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.

Contents

Preliminaries

The usual way of stating the axioms presumes a two sorted first order language L ∗ with a single binary relation symbol ∈ . 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 ∈ , while those for urelements may only appear on the left, i.e. the following are examples of valid expressions: p ∈ a , b ∈ 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, ∈ , ¬ , ∧ , ∨ , and bounded quantification. That is quantification of the form ∀ x ∈ a or ∃ x ∈ a where a is given set.

Axioms

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

  • Extensionality: ∀ x ( x ∈ a ↔ x ∈ b ) → a = b
  • Foundation: This is an axiom schema where for every formula ϕ ( x ) we have ∃ a . ϕ ( a ) → ∃ a ( ϕ ( a ) ∧ ∀ x ∈ a ( ¬ ϕ ( x ) ) ) .
  • Pairing: ∃ a ( x ∈ a ∧ y ∈ a )
  • Union: ∃ a ∀ c ∈ b . ∀ y ∈ c ( y ∈ a )
  • Δ0-Separation: This is again an axiom schema, where for every Δ 0 -formula ϕ ( x ) we have the following ∃ a ∀ x ( x ∈ a ↔ x ∈ b ∧ ϕ ( x ) ) .
  • Δ 0 -Collection: This is also an axiom schema, for every Δ 0 -formula ϕ ( x , y ) we have ∀ x ∈ a . ∃ y . ϕ ( x , y ) → ∃ b ∀ x ∈ a . ∃ y ∈ b . ϕ ( x , y ) .
  • Set Existence: ∃ a ( a = a )

    • Additional assumptions

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

  • ∀ p ∀ a ( p ≠ a )
  • ∀ p ∀ x ( x ∉ 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.

    References

    Kripke–Platek set theory with urelements Wikipedia
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