Kripke–Platek set theory

E387803

axiomatic set theory set theory subsystem of Zermelo–Fraenkel set theory

Kripke–Platek set theory is a weaker, predicative subsystem of Zermelo–Fraenkel set theory focused on sets that are explicitly constructible and often used in the study of admissible sets and recursion theory.

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All labels observed (5)

Label	Occurrences
Kripke–Platek set theory with urelements	2
Admissible Sets and Structures	1
Kripke–Platek set theory canonical	1
Kripke–Platek set theory with Infinity	1
Kripke–Platek set theory without Infinity	1

How this entity was disambiguated

This entity first appeared as the object of triple T3780712 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Kripke–Platek set theory
Context triple: [set theory, hasAxiomSystem, Kripke–Platek set theory]

A. von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
B. Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
C. Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
D. Morse–Kelley set theory by class–set distinction
Morse–Kelley set theory by class–set distinction is a foundational system that avoids certain set-theoretic paradoxes by rigorously distinguishing between sets and proper classes within a powerful axiomatic framework.
E. Foundations of Set Theory (with Andrey Kolmogorov)
"Foundations of Set Theory" is a classic 20th-century mathematical text co-authored by Pavel Alexandrov and Andrey Kolmogorov that systematically develops the basic concepts and axioms of set theory.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Kripke–Platek set theory
Target entity description: Kripke–Platek set theory is a weaker, predicative subsystem of Zermelo–Fraenkel set theory focused on sets that are explicitly constructible and often used in the study of admissible sets and recursion theory.

A. von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
B. Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
C. Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
D. Morse–Kelley set theory by class–set distinction
Morse–Kelley set theory by class–set distinction is a foundational system that avoids certain set-theoretic paradoxes by rigorously distinguishing between sets and proper classes within a powerful axiomatic framework.
E. Foundations of Set Theory (with Andrey Kolmogorov)
"Foundations of Set Theory" is a classic 20th-century mathematical text co-authored by Pavel Alexandrov and Andrey Kolmogorov that systematically develops the basic concepts and axioms of set theory.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	axiomatic set theory ⓘ set theory ⓘ subsystem of Zermelo–Fraenkel set theory ⓘ
abbreviation	KP ⓘ
assumes	regularity of membership relation ⓘ
characterizes	admissible sets ⓘ
developedIn	20th century ⓘ
field	mathematical logic ⓘ
focusesOn	predicative aspects of set theory ⓘ
formalizes	recursion on admissible ordinals ⓘ
hasAxiom	Extensionality ⓘ Foundation ⓘ Infinity ⓘ Pairing ⓘ Union ⓘ Δ0-Collection ⓘ Δ0-Separation ⓘ
hasConsequence	every set is well-founded ⓘ
hasConservativeExtension	Kripke–Platek set theory with urelements ⓘ linked to: Kripke–Platek set theory
hasModel	every admissible set ⓘ
hasProofTheoreticOrdinal	Bachmann–Howard ordinal ⓘ
hasVariant	Kripke–Platek set theory with Infinity ⓘ linked to: Kripke–Platek set theory Kripke–Platek set theory with urelements ⓘ linked to: Kripke–Platek set theory Kripke–Platek set theory without Infinity ⓘ linked to: Kripke–Platek set theory
implies	basic arithmetic truths ⓘ
isInterpretableIn	Zermelo–Fraenkel set theory ⓘ
isPredicative	true ⓘ
isSubsystemOf	first-order logic with equality ⓘ
isWeakerThan	Peano arithmetic plus certain transfinite induction principles ⓘ
languageIncludes	equality = ⓘ membership relation ∈ ⓘ
namedAfter	Richard Platek ⓘ Saul Kripke ⓘ
oftenComparedWith	Zermelo set theory ⓘ constructive Zermelo–Fraenkel set theory ⓘ linked to: constructive set theory
omitsAxiom	Power set axiom ⓘ Replacement schema ⓘ full Separation schema ⓘ
relatedTo	Lα (levels of the constructible universe) ⓘ admissible ordinals ⓘ constructible hierarchy ⓘ
studiedIn	ordinal analysis ⓘ subsystems of second-order arithmetic ⓘ
supports	Δ0-recursion ⓘ
usedIn	constructive set theory ⓘ proof theory ⓘ recursion theory ⓘ theory of admissible sets ⓘ
weakerThan	Zermelo–Fraenkel set theory ⓘ

How these facts were elicited