Kripke–Platek set theory

Kripke–Platek set theory
Name	Kripke–Platek set theory
Abbreviation	KP
Introduced	1960s
Introduced by	Saul Kripke; Richard Platek
Field	Mathematical logic; Set theory
Notable for	Foundations of admissible sets; recursion-theoretic analysis

Kripke–Platek set theory is an axiomatic system for set theory emphasizing predicative comprehension and admissibility, developed in the 1960s by Saul Kripke and Richard Platek and used in investigations connected to recursion theory, constructibility, and proof-theoretic ordinal analysis. It plays a central role in work by Dana Scott, Michael Rathjen, John Steel, and Kurt Gödel on models of set theory, and interacts with research from Alfred Tarski, Solomon Feferman, and Stephen Simpson on definability, recursion, and foundations. KP is weaker than Zermelo–Fraenkel set theory and often appears alongside studies by Gerhard Gentzen, Georg Kreisel, and W. Hugh Woodin concerning consistency strength, inner models, and determinacy.

Overview

KP focuses on sets that are "admissible" in the sense pioneered by Dana Scott and furthered by John Barwise, and it omits power set and full separation to capture a minimal framework used in analyses by Kurt Gödel, Paul Cohen, and Ronald Jensen of constructible hierarchies and inner models. The theory has been central to work by Michael Rathjen, Jon Barwise, Azriel Lévy, and H. Jerome Keisler on infinitary languages, definability, and recursion-theoretic hierarchies, and it figures in studies by Gerald Sacks, Harvey Friedman, and Richard Shore regarding hyperarithmetical theory, Π^1_1-comprehension, and reverse mathematics. KP's development influenced research directions taken by Alan Turing, Kleene, and Emil Post in computability, and its admissible set apparatus has been applied in investigations by Hugh Woodin, William Mitchell, and John Steel of determinacy and inner model theory.

Axioms and Formal System

The KP axioms include extensionality, foundation, pairing, union, infinity, Δ0-separation, and Δ0-collection (also called Σ-collection in some variants), presented in a first-order language much as in the expositions by Alfred Tarski, Solomon Feferman, and Dana Scott; Kurt Gödel's work on constructible sets provided motivation for the restricted comprehension principles. The role of Δ0-formulas connects KP to techniques used by Kleene, Stephen Kleene, and Gerald Sacks in recursion theory, while Δ0-collection and foundation relate to principles studied by Georg Cantor, Ernst Zermelo, and Abraham Fraenkel in early set-theoretic axiomatizations. Variants drop or modify infinity or add Power Set analogues, as considered in the literature of Paul Cohen, Ronald Jensen, and John Conway when comparing consistency strengths to Zermelo–Fraenkel and ZF + Choice frameworks.

Models and Relative Consistency

Models of KP are often constructed via relativized constructible hierarchies Lα and admissible ordinals investigated by Kurt Gödel, Ronald Jensen, and Dana Scott; such constructions connect to work by Gödel on constructibility and Cohen on forcing and independence. Relative consistency proofs for KP or extensions employ techniques mirrored in studies by Gerhard Gentzen, Solomon Feferman, and Kurt Gödel involving proof-theoretic reductions and inner model comparisons analyzed by Hugh Woodin, William Mitchell, and John Steel. Admissible sets and their minimality properties are central in analyses by Jon Barwise, Michael Rathjen, and Azriel Lévy, and they link to ordinal analyses carried out by Wilfried Buchholz, Wolfram Pohlers, and Michael Rathjen for subsystems of arithmetic and set theory.

Constructible Hierarchy and Admissible Sets

The constructible hierarchy L and admissible ordinals α underpin the canonical models of KP, with foundational contributions by Kurt Gödel, Ronald Jensen, and Dana Scott; studies by Gerald Sacks, Harvey Friedman, and Azriel Lévy examine how admissible sets reflect recursion-theoretic and definability properties. The notion of admissibility connects KP to descriptive set theory research by John Steel, Donald A. Martin, and Hugh Woodin on determinacy, and to inner model theoretic constructions of William Mitchell, Ronald Jensen, and Sol Feferman that compare large cardinal hypotheses to admissible-level phenomena. Results about Σ1-definability, projecta, and condensation in admissible structures are often attributed to Jensen, Barwise, and Scott and have consequences for recursion theorists such as Stephen Simpson and Gerald Sacks.

Proof Theory and Computability Connections

Proof-theoretic ordinals for KP and its variants have been analyzed by Gerhard Gentzen, Michael Rathjen, Wilfried Buchholz, and Wolfram Pohlers, linking ordinal analysis to consistency statements and proof-theoretic reductions developed in the work of Kurt Gödel, Gerhard Gentzen, and Georg Kreisel. KP's alignment with admissible recursion means techniques from Alan Turing, Stephen Kleene, Emil Post, and Gerald Sacks inform its computational content, while reverse-mathematics style comparisons performed by Harvey Friedman, Stephen Simpson, and Richard Shore situate KP among subsystems studied in foundations of mathematics. Connections to higher-type recursion and ordinal-recursive functions leverage methods from Solomon Feferman, William W. Tait, and Michael Rathjen.

Variants and Extensions

Several variants of KP modify comprehension, collection, or choice principles, as discussed by Ronald Jensen, Azriel Lévy, and Paul Cohen, and extensions that incorporate replacement, full separation, or large cardinal assumptions are considered in the literature of Kurt Gödel, John Steel, Hugh Woodin, and William Mitchell. Theories like KPω, KP(P), and KP + Σn-Collection appear in comparative studies by Solomon Feferman, Michael Rathjen, and Harvey Friedman relating to Π^1_1-comprehension, admissible recursion, and determinacy results by Donald A. Martin and John Steel. Applications and further developments are surveyed in works by Jon Barwise, Dana Scott, and Ronald Jensen, who connect KP variants to descriptive set theory, inner model theory, and proof theory.

Category:Set theory