Hilbert's first problem
E628900
Hilbert's first problem is one of David Hilbert’s famous list of 23 problems, asking whether there exists a set whose size is strictly between that of the integers and the real numbers, i.e., the status of the continuum hypothesis.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf | mathematical problem ⓘ |
| asksWhether |
there exists a set of cardinality strictly between the integers and the real numbers
ⓘ
there exists a set whose cardinality is strictly between aleph-null and the cardinality of the continuum ⓘ |
| concerns |
cardinality of infinite sets
ⓘ
continuum hypothesis NERFINISHED ⓘ foundations of mathematics ⓘ set theory ⓘ |
| equivalentToQuestion | Is the continuum hypothesis true? ⓘ |
| firstPublishedIn | 1900 ⓘ |
| formalContext |
Zermelo–Fraenkel set theory
NERFINISHED
ⓘ
Zermelo–Fraenkel set theory with Choice NERFINISHED ⓘ |
| hasPhilosophicalAspect |
completeness of axiomatic systems
ⓘ
nature of mathematical infinity ⓘ |
| hasSolutionType | independence result ⓘ |
| hasStandardFormulation | Is there a set whose cardinality is strictly between that of the integers and that of the real numbers? ⓘ |
| influencedField |
foundations of mathematics
ⓘ
mathematical logic ⓘ set theory ⓘ |
| involvesConcept |
aleph-null
ⓘ
cardinality of the continuum ⓘ integers ⓘ real numbers ⓘ set-theoretic independence ⓘ uncountable sets ⓘ |
| languageOfOriginalStatement | German ⓘ |
| namedAfter | David Hilbert NERFINISHED ⓘ |
| numberInHilbertList | 1 ⓘ |
| originalPublication | Mathematische Probleme (Hilbert's 1900 address) NERFINISHED ⓘ |
| partOf | Hilbert's list of 23 problems NERFINISHED ⓘ |
| presentedAt | International Congress of Mathematicians 1900 NERFINISHED ⓘ |
| presentedInCity | Paris NERFINISHED ⓘ |
| relatedTo |
Cantor's continuum hypothesis
NERFINISHED
ⓘ
generalized continuum hypothesis NERFINISHED ⓘ |
| resolutionYearPartial | 1938 ⓘ |
| resolutionYearPartial |
1940
ⓘ
1963 ⓘ |
| resolvedBy |
Kurt Gödel
NERFINISHED
ⓘ
Paul Cohen NERFINISHED ⓘ |
| resultByCohen |
continuum hypothesis is independent of ZF if ZF is consistent
NERFINISHED
ⓘ
continuum hypothesis is independent of ZFC if ZFC is consistent ⓘ |
| resultByGödel | continuum hypothesis is consistent with ZF if ZF is consistent ⓘ |
| resultByGödel | continuum hypothesis is consistent with ZFC if ZFC is consistent ⓘ |
| statedBy | David Hilbert NERFINISHED ⓘ |
| statusInZFC | independent ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.