constructible universe

E100621

class model of set theory inner model proper class

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.

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All labels observed (3)

Label	Occurrences
Consistency of CH	1
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	Gödel constructible universe ⓘ linked to: 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 ⓘ ZFC ⓘ linked to: ZF
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

Referenced by (3)

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

Kurt Gödel → notableWork → constructible universe ⓘ

constructible universe → hasAlternativeName → Gödel constructible universe ⓘ

linked to: constructible universe

The Consistency of the Continuum Hypothesis → hasAbbreviation → Consistency of CH ⓘ

subject linked to: Gödel 1940 monograph "The Consistency of the Continuum Hypothesis"