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 (2)
| Label | Occurrences |
|---|---|
| continuum hypothesis canonical | 5 |
| Hypothèse du continu | 1 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical hypothesis
ⓘ
set-theoretic hypothesis ⓘ |
| canBeAssumedAsAxiom | yes ⓘ |
| canBeNegatedAsAxiom | yes ⓘ |
| consequenceOf | generalized continuum hypothesis ⓘ |
| dateProposed | late 19th century ⓘ |
| discussedIn |
Hilbert problems
ⓘ
surface form:
Hilbert's 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 |
ZF
ⓘ
surface form:
ZFC
Zermelo–Fraenkel set theory ⓘ
surface form:
Zermelo–Fraenkel set theory with the axiom of choice
|
| 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 |
ZF
ⓘ
surface form:
ZFC
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 ⓘ |
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hypothèse du continu