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The Kripke-Platek axioms of set theory (KP) are a system of axioms of axiomatic set theory. The axiom system is written in first-order logic; it has an infinite number of axioms because an axiom schema is used.
The axioms of KP are:
- Axiom of extensionality: Two sets are the same if and only if they have the same elements.
- 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 induction: If φ(a) is a proposition, 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 infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y ∪ {y}.
- Axiom of cartesic product: If x and y are sets, then there is a set z consisting of all pairs (a, b) of elements a of x and b of y.
- Axiom of Σ0-separation: Given any set and any Σ0-proposition 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-proposition φ(y, x), then, if for every set x there exists a set y such that φ(y, x) holds, then for all sets u there exists a set v such that for every x ∈ u there is a y ∈ v such that φ(y, x) holds.
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