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 |
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gödel constructible universe