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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (4)

Label	Occurrences
Kripke–Platek set theory with urelements	2
Kripke–Platek set theory canonical	1
Kripke–Platek set theory with Infinity	1
Kripke–Platek set theory without Infinity	1

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 self-linksurface differs ⓘ surface form: Kripke–Platek set theory with urelements
hasModel	every admissible set ⓘ
hasProofTheoreticOrdinal	Bachmann–Howard ordinal ⓘ
hasVariant	Kripke–Platek set theory self-linksurface differs ⓘ surface form: Kripke–Platek set theory with Infinity Kripke–Platek set theory self-linksurface differs ⓘ surface form: Kripke–Platek set theory with urelements Kripke–Platek set theory self-linksurface differs ⓘ surface form: Kripke–Platek set theory without Infinity
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 set theory ⓘ surface form: constructive Zermelo–Fraenkel 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 ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

set theory → hasAxiomSystem → Kripke–Platek set theory ⓘ

Kripke–Platek set theory → hasVariant → Kripke–Platek set theory self-linksurface differs ⓘ

this entity surface form: Kripke–Platek set theory with urelements

Kripke–Platek set theory → hasVariant → Kripke–Platek set theory self-linksurface differs ⓘ

this entity surface form: Kripke–Platek set theory with Infinity

Kripke–Platek set theory → hasVariant → Kripke–Platek set theory self-linksurface differs ⓘ

this entity surface form: Kripke–Platek set theory without Infinity

Kripke–Platek set theory → hasConservativeExtension → Kripke–Platek set theory self-linksurface differs ⓘ

this entity surface form: Kripke–Platek set theory with urelements