constructible universe
E100621
The constructible universe is a class model of set theory introduced by Kurt Gödel that systematically builds sets in hierarchical stages and shows the relative consistency of the axiom of choice and the generalized continuum hypothesis with ZF.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gödel constructible universe | 1 |
| constructible universe canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T839946 — 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: constructible universe Context triple: [Kurt Gödel, notableWork, constructible universe]
-
A.
Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
-
B.
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.
-
C.
von Neumann universe
The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.
-
D.
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.
-
E.
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.
- 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: constructible universe Target entity description: The constructible universe is a class model of set theory introduced by Kurt Gödel that systematically builds sets in hierarchical stages and shows the relative consistency of the axiom of choice and the generalized continuum hypothesis with ZF.
-
A.
Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
-
B.
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.
-
C.
von Neumann universe
The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
class model of set theory
ⓘ
inner model ⓘ proper class ⓘ |
| builtInStagesIndexedBy | ordinals ⓘ |
| constructionMethod | definability over earlier stages ⓘ |
| containsAllOrdinals | true ⓘ |
| hasAlternativeName |
constructible universe
ⓘ
surface form:
Gödel constructible universe
|
| hasConsequence |
no large cardinals beyond certain small ones if V = L
ⓘ
no measurable cardinals if V = L ⓘ |
| hasProperty |
every set is constructible
ⓘ
every set is ordinal definable in L ⓘ minimal inner model of ZFC ⓘ well-ordered by a definable global well-order ⓘ |
| hasReferenceWork | Gödel 1940 monograph "The Consistency of the Continuum Hypothesis" ⓘ |
| hasStage | L_alpha ⓘ |
| hasSymbol | L ⓘ |
| impliesStatement | V = L ⓘ |
| introducedBy | Kurt Gödel ⓘ |
| introducedInContextOf | relative consistency proofs ⓘ |
| introducedInYear | 1938 ⓘ |
| isContainedIn | V ⓘ |
| isDefinedAs | union over all ordinals of L_alpha ⓘ |
| isStudiedIn |
mathematical logic
ⓘ
set theory ⓘ |
| isSubsetOf | von Neumann universe ⓘ |
| isToolIn |
descriptive set theory
ⓘ
inner model theory ⓘ proof theory of set theory ⓘ |
| isTransitiveClass | true ⓘ |
| relatedPrinciple | V = L axiom ⓘ |
| satisfiesAxiom |
axiom of choice
ⓘ
axiom of extensionality ⓘ axiom of foundation ⓘ axiom of infinity ⓘ axiom of pairing ⓘ axiom of power set ⓘ axiom of replacement ⓘ axiom of union ⓘ axiom schema of replacement ⓘ axiom schema of separation ⓘ |
| satisfiesStatement | generalized continuum hypothesis ⓘ |
| satisfiesTheory |
ZF
ⓘ
ZF ⓘ
surface form:
ZFC
|
| stage0Equals | empty set ⓘ |
| stageLimitDefinition | L_lambda = union_{alpha<lambda} L_alpha for limit lambda ⓘ |
| stageSuccessorDefinition | L_{alpha+1} = Def(L_alpha) ⓘ |
| usedToShow |
relative consistency of the axiom of choice with ZF
ⓘ
relative consistency of the generalized continuum hypothesis with ZF ⓘ |
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: constructible universe Description of subject: The constructible universe is a class model of set theory introduced by Kurt Gödel that systematically builds sets in hierarchical stages and shows the relative consistency of the axiom of choice and the generalized continuum hypothesis with ZF.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gödel constructible universe