continuum hypothesis
E160402
The continuum hypothesis is a central conjecture in set theory proposing a specific relationship between the sizes of the set of real numbers and the set of natural numbers, famously shown to be independent of the standard axioms of mathematics.
All labels observed (5)
| Label | Occurrences |
|---|---|
| continuum hypothesis canonical | 7 |
| Cantor's continuum hypothesis | 1 |
| Continuum Hypothesis | 1 |
| Hypothèse du continu | 1 |
| continuum hypothesis is independent of ZF if ZF is consistent | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1396011 — 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: continuum hypothesis Context triple: [Georg Cantor, knownFor, continuum hypothesis]
-
A.
constructible universe
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.
-
B.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
-
C.
Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
-
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.
set theory
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
- 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: continuum hypothesis Target entity description: The continuum hypothesis is a central conjecture in set theory proposing a specific relationship between the sizes of the set of real numbers and the set of natural numbers, famously shown to be independent of the standard axioms of mathematics.
-
A.
constructible universe
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.
-
B.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
-
C.
Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
-
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.
set theory
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical hypothesis
ⓘ
set-theoretic hypothesis ⓘ |
| canBeAssumedAsAxiom | yes ⓘ |
| canBeNegatedAsAxiom | yes ⓘ |
| consequenceOf | generalized continuum hypothesis ⓘ |
| dateProposed | late 19th century ⓘ |
| discussedIn |
Hilbert's problems
ⓘ
linked to:
Hilbert problems
|
| equivalentFormulation | 2^{ℵ₀} = ℵ₁ ⓘ |
| field | set theory ⓘ |
| formalStatement | There is no set A such that |ℕ| < |A| < |ℝ| ⓘ |
| hasPhilosophicalAspect |
debate about truth in mathematics
ⓘ
discussion of maximality principles in set theory ⓘ |
| implies |
Every infinite subset of ℝ has cardinality ℵ₀ or 2^{ℵ₀}
ⓘ
The first uncountable cardinal equals the cardinality of the continuum ⓘ |
| independenceFrom |
ZFC
ⓘ
linked to:
ZF
Zermelo–Fraenkel set theory with the axiom of choice ⓘ
linked to:
Zermelo–Fraenkel set theory
|
| independenceResultPart |
Cohen showed CH cannot be proved from ZFC if ZFC is consistent
ⓘ
Gödel showed CH cannot be disproved from ZFC if ZFC is consistent ⓘ |
| influenced |
development of modern set theory
ⓘ
research on determinacy axioms ⓘ research on large cardinals ⓘ |
| involvesConcept |
aleph numbers
ⓘ
cardinality ⓘ continuum ⓘ continuum cardinality ⓘ power set ⓘ well-ordering of the reals ⓘ |
| involvesSet |
set of natural numbers ℕ
ⓘ
set of real numbers ℝ ⓘ |
| mainTopic |
cardinality of the continuum
ⓘ
cardinality of the real numbers ⓘ infinite cardinals ⓘ |
| methodUsedInIndependenceProof |
constructible universe L
ⓘ
forcing ⓘ |
| openQuestion | Whether CH is true in an absolute sense beyond ZFC ⓘ |
| positionInHilbertProblems | Hilbert's first problem ⓘ |
| proposedBy | Georg Cantor ⓘ |
| relatedTo |
ZFC
ⓘ
linked to:
ZF
Zermelo–Fraenkel set theory ⓘ axiom of choice ⓘ generalized continuum hypothesis ⓘ |
| shownIndependentBy |
Kurt Gödel
ⓘ
Paul Cohen ⓘ |
| states | There is no set whose cardinality is strictly between that of the integers and the real numbers ⓘ |
| statusInZFC | independent ⓘ |
| symbol | CH ⓘ |
| yearOfCohenResult | 1963 ⓘ |
| yearOfGödelResult | 1940 ⓘ |
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: continuum hypothesis Description of subject: The continuum hypothesis is a central conjecture in set theory proposing a specific relationship between the sizes of the set of real numbers and the set of natural numbers, famously shown to be independent of the standard axioms of mathematics.
Referenced by (11)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: continuum hypothesis
subject linked to:
Gödel 1940 monograph "The Consistency of the Continuum Hypothesis"
linked to: continuum hypothesis
linked to: continuum hypothesis
Hilbert's first problem
→
resultByCohen
→
continuum hypothesis is independent of ZF if ZF is consistent
ⓘ
linked to: continuum hypothesis