Kripke–Platek set theory

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The Kripke–Platek axioms of set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, are a system of axiomatic set theory based on the ideas of Saul Kripke (1964) and Richard Platek (1966).

KP is weaker than Zermelo–Fraenkel set theory (ZFC). Unlike ZFC, KP does not include the power set 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[edit]

  • Axiom of extensionality: Two sets are the same if and only if they have the same elements.
  • Axiom of induction: φ(a) being a formula, if for all sets x - the assumption that φ(y) holds for all elements y of x - entails 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[1] 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 \exist u \in v. (More generally, we would say that a formula is Σn+1 when it is obtained by adding existential quantifiers in front of a Πn formula, and that it is Πn+1 when it is obtained by adding universal quantifiers in front of a Σn 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 in KP 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[edit]

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.

Proof: {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: \exist r \in p (a \in r \and \forall x \in r (x = a)) \and \exist s \in p (a \in s \and b \in s \and \forall x \in s (x = a \or x = b)) and \forall t \in p ((a \in t \and \forall x \in t (x = a)) \or (a \in t \and b \in t \and \forall x \in t (x = a \or 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 \exist 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 \exist p \in v \psi (a, b, p) \and \forall p \in v \exist a \in A \psi (a, b, p).

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

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

Finally, A×B = \cup{A×{b} | b in B} exists by the axiom of union. QED.

Admissible sets[edit]

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 Lα 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(Lα) 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 Lα 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[edit]

References[edit]

  • Gostanian, Richard (1980). "Constructible Models of Subsystems of ZF". Journal of Symbolic Logic (Association for Symbolic Logic) 45 (2): 237. doi:10.2307/2273185. JSTOR 2273185. 
  • Kripke, S. (1964), "Transfinite recursion on admissible ordinals", J. Symbolic logic 29: 161–162 
  • Platek, Richard Alan (1966), Foundations of recursion theory, Thesis (Ph.D.)–Stanford University, MR 2615453 
  1. ^ Poizat, Bruno (2000). A course in model theory: an introduction to contemporary mathematical logic. Springer. ISBN 0-387-98655-3. , note at end of §2.3 on page 27: “Those who do not allow relations on an empty universe consider (∃x)x=x and its consequences as theses; we, however, do not share this abhorrence, with so little logical ground, of a vacuum.”