Kripke–Platek set theory

The Kripke–Platek axioms of set theory (KP) (Template:Pron-en) are a system of axioms for axiomatic set theory developed by Saul Kripke and Richard Platek. The axiom system, written in first-order logic, has an infinite number of axioms because an infinite axiom schema is used.

KP is weaker than Zermelo–Fraenkel set theory (ZFC). Unlike ZFC, KP does not include the powerset axiom, and KP includes only limited forms of the axiom of separation and axiom of replacement from ZFC. These restrictions on the axioms of KP lead to close connections between KP, generalized recursion theory, and the theory of admissible ordinals.

The axioms of KP

• Axiom of extensionality: Two sets are the same if and only if they have the same elements.
• Axiom of induction: If φ(a) is a formula, and if for all sets x it follows from the fact that φ(y) is true for all elements y of x that φ(x) holds, then φ(x) holds for all sets x.
• Axiom of empty set: There exists a set with no members, called the empty set and denoted {}. (Note: the existence of a member in the universe of discourse, i.e., ∃x(x=x), is implied in certain formulations of first-order logic, in which case the axiom of empty set follows from the axiom of separation, and is thus redundant.)
• Axiom of pairing: If x, y are sets, then so is {x, y}, a set containing x and y as its only elements.
• Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
• Axiom of Σ₀-separation: Given any set and any Σ₀-formula φ(x), there is a subset of the original set containing precisely those elements x for which φ(x) holds. (This is an axiom schema.)
• Axiom of Σ₀-collection: Given any Σ₀-formula φ(x, y), if for every set x there exists a set y such that φ(x, y) holds, then for all sets u there exists a set v such that for every x in u there is a y in v such that φ(x, y) holds.

Here, a Σ₀, or Π₀, or Δ₀ formula is one all of whose quantifiers are bounded. This means any quantification is the form $\forall u\in v$ or $\exists u\in v.$ (More generally, we would say that a formula is Σ_n+1 when it is obtained by adding existential quantifiers in front of a Π_n formula, and that it is Π_n+1 when it is obtained by adding universal quantifiers in front of a Σ_n formula: this is related to the arithmetical hierarchy but in the context of set theory.)

These axioms differ from ZFC in as much as they exclude the axioms of: infinity, powerset, and choice. Also the axioms of separation and collection here are weaker than the corresponding axioms in ZFC because the formulas φ used in these are limited to bounded quantifiers only.

The axiom of induction here is stronger than the usual axiom of regularity (which amounts to applying induction to the complement of a set (the class of all sets not in the given set)).

Proof that Cartesian products exist

Theorem: If A and B are sets, then there is a set A×B which consists of all ordered pairs (a, b) of elements a of A and b of B.

{a} = {a, a} exists by the axiom of pairing. {a, b} exists by the axiom of pairing. Thus (a, b) = { {a}, {a, b} } exists by the axiom of pairing.

If p is intended to stand for (a, b), then a Δ₀ formula expressing that is: $\exists r\in p(a\in r\land \forall x\in r(x=a))\land \exists s\in p(a\in s\land b\in s\land \forall x\in s(x=a\lor x=b))$ and $\forall t\in p((a\in t\land \forall x\in t(x=a))\lor (a\in t\land b\in t\land \forall x\in t(x=a\lor x=b))).$

Thus a superset of A×{b} = {(a, b) | a in A} exists by the axiom of collection.

Abbreviate the formula above by $\psi (a,b,p)\!.$ Then $\exists a\in A\psi (a,b,p)$ is Δ₀. Thus A×{b} itself exists by the axiom of separation.

If v is intended to stand for A×{b}, then a Δ₀ formula expressing that is: $\forall a\in A\exists p\in v\psi (a,b,p)\land \forall p\in v\exists a\in A\psi (a,b,p).$

Thus a superset of {A×{b} | b in B} exists by the axiom of collection.

Putting $\exists b\in B$ in front of that last formula and we get that the set {A×{b} | b in B} itself exists by the axiom of separation.

Finally, A×B = $\cup${A×{b} | b in B} exists by the axiom of union. This is what was to be proved.

Admissible sets

A set $A\,$ is called admissible if it is transitive and $\langle A,\in \rangle$ is a model of Kripke–Platek set theory.

An ordinal number α is called an admissible ordinal if L_α is an admissible set.

The ordinal α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ<α for which there is a Σ₁(L_α) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.

If L_α is a standard model of KP set theory without the axiom of Σ₀-collection, then it is said to be an "amenable set".

See also

• Constructible universe
• Admissible ordinal
• Kripke–Platek set theory with urelements

References

• Gostanian, Richard (1980), "Constructible Models of Subsystems of ZF", Journal of Symbolic Logic, 45 (2): 237, doi:10.2307/2273185