The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it. Axioms In its formulation, a Δ0 formula is one all of whose quantifiers are bounded. This means any quantification is the form or (See the Lévy hierarchy.) 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 {}. 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 Δ0-separation: Given any set and any Δ0 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 Δ0-collection: Given any Δ0 formula φ(x, y), if for every set x there exists a set y such that φ(x, y) holds, then for all sets X there exists a set Y such that for every x in X there is a y in Y such that φ(x, y) holds. Some but not all authors include an Axiom of infinity KP with infinity is denoted by KPω. These axioms lead to close connections between KP, generalized recursion theory, and the theory of admissible ordinals. KP can be studied as a constructive set theory by dropping the law of excluded middle, without changing any axioms. Empty set If any set is postulated to exist, such as in the axiom of infinity, then the axiom of empty set is redundant because it is equal to the subset . Furthermore, 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 Δ0-separation, and is thus redundant. Comparison with Zermelo-Fraenkel set theory As