Aleph (mathematics)
E622067
Aleph (mathematics) denotes the infinite cardinal numbers used in set theory to measure and compare the sizes of infinite sets.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
cardinal number symbol
ⓘ
mathematical notation ⓘ |
| alphabetOrigin | Hebrew letter aleph (ℵ) NERFINISHED ⓘ |
| appearsIn |
Zermelo–Fraenkel set theory
NERFINISHED
ⓘ
von Neumann–Bernays–Gödel set theory NERFINISHED ⓘ |
| assumes | axiom of choice (for well-ordering all sets) ⓘ |
| cardinalityExample |
|countable infinite set| = ℵ₀
ⓘ
|ℕ| = ℵ₀ ⓘ |
| cardinalityType | transfinite cardinal ⓘ |
| category |
mathematical symbol
ⓘ
set-theoretic concept ⓘ |
| contrastedWith |
beth numbers
NERFINISHED
ⓘ
ordinal numbers ⓘ |
| denotes | infinite cardinal numbers ⓘ |
| dependsOn | ordinal indexing α ⓘ |
| distinguishes | different infinite cardinalities ⓘ |
| domain | cardinal numbers ⓘ |
| field | set theory ⓘ |
| firstElement |
aleph-null
ⓘ
ℵ₀ NERFINISHED ⓘ |
| firstElementOfType | smallest infinite cardinal ⓘ |
| formalization | ℵ_α is the α-th infinite cardinal ⓘ |
| generalElementNotation | ℵ_α ⓘ |
| historicalPeriod | late 19th century mathematics ⓘ |
| introducedBy | Georg Cantor NERFINISHED ⓘ |
| namingConvention | ℵ₀, ℵ₁, ℵ₂, … ⓘ |
| notationFor | well-ordered infinite cardinals ⓘ |
| notationType | subscripted family of symbols ⓘ |
| notUsedFor | finite cardinals ⓘ |
| parameterizedBy | ordinal numbers ⓘ |
| relatedConcept |
cardinality of the natural numbers
ⓘ
cardinality of the real numbers ⓘ continuum hypothesis ⓘ generalized continuum hypothesis ⓘ |
| requires | well-ordering of sets to define sequence ℵ_α ⓘ |
| secondElement |
aleph-one
ⓘ
ℵ₁ ⓘ |
| standardIn | modern set theory textbooks ⓘ |
| symbolForm | ℵ ⓘ |
| usedFor |
comparing sizes of infinite sets
ⓘ
measuring sizes of infinite sets ⓘ |
| usedIn |
axiomatic set theory
ⓘ
cardinal arithmetic ⓘ transfinite numbers ⓘ |
| usedToExpress | hierarchy of infinite sizes ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.