The Kripke–Platek set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, 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.
In its formulation, a Δ0 formula is one all of whose quantifiers are bounded.
Some but not all authors include an 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.
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[1] of first-order logic, in which case the axiom of empty set follows from the axiom of Δ0-separation, and is thus redundant.
As noted, the above are weaker than ZFC as they exclude the power set axiom, choice, and sometimes infinity.
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 the context of 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: The singleton set with member a, written {a}, is the same as the unordered pair {a, a}, by the axiom of extensionality.
expressing that p stands for the pair (a, b) is given by the lengthy What follows are two steps of collection of sets, followed by a restriction through separation.
All results are also expressed using set builder notation.
KP fails to prove some common theorems in set theory, such as the Mostowski collapse lemma.