KripkePlatek

Kripke–Platek set theory (KP) is a weak foundational system in mathematical logic and set theory, introduced by Saul Kripke and Richard Platek in the 1960s to study admissible sets and predicative constructions. It is formulated in the language of set membership and is considerably weaker than Zermelo–Fraenkel set theory (ZF); it omits the Powerset and full Replacement schemes and employs restricted forms of comprehension.

The standard axioms of KP include Extensionality, the existence of the empty set, pairing, union, and an axiom of Infinity to guarantee an infinite universe. In addition, KP uses restricted schemas: Delta_0-Separation, which allows forming subsets only by Delta_0-definable properties, and Delta_0-Collection, which ensures that if each element of a set has a Delta_0-definable witness, then there is a set containing all such witnesses. Variants differ in whether Infinity is assumed and in the exact strength of the collection or separation schemes used.

KP is central to the study of admissible sets: transitive models that satisfy KP and are closed under Delta_0-definable operations. An admissible set can be viewed as a robust, self-contained universe suitable for conducting recursion-theoretic and definability analyses. Because it avoids large parts of ZF, KP provides a more malleable framework for exploring foundational questions and the interactions between computability, logic, and set theory.

Variants of KP exist, including KP without Infinity and KP with different formulations of the collection and separation axioms. KP and its variants continue to influence areas such as inner model theory, constructive set theory, and descriptive set theory, where a weaker, well-behaved setting is advantageous.