Kripke–Platek set theory
E387803
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.
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 |
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
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Kripke–Platek set theory Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kripke–Platek set theory with urelements
this entity surface form:
Kripke–Platek set theory with Infinity
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