Bernstein theorem
E628898
Bernstein theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bernstein theorem canonical | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
set theory theorem
ⓘ
theorem ⓘ |
| alsoKnownAs |
Cantor–Bernstein theorem
NERFINISHED
ⓘ
Cantor–Bernstein–Schröder theorem NERFINISHED ⓘ Schröder–Bernstein theorem NERFINISHED ⓘ |
| appliesTo |
finite sets
ⓘ
infinite sets ⓘ |
| clarifies | relationship between injections and bijections ⓘ |
| concerns |
comparisons of cardinalities
ⓘ
existence of bijections ⓘ pairs of sets ⓘ |
| doesNotRequire | axiom of choice ⓘ |
| ensures | existence of a bijection under mutual embeddability of sets ⓘ |
| field |
mathematics
ⓘ
set theory ⓘ |
| formalization | If |A| ≤ |B| and |B| ≤ |A| then |A| = |B|. ⓘ |
| generalizes | pigeonhole principle for cardinalities ⓘ |
| gives | criterion for equality of cardinalities ⓘ |
| hasProofMethod |
construction of a bijection from two injections
ⓘ
set-theoretic decomposition of domains ⓘ |
| holdsIn | Zermelo–Fraenkel set theory without the axiom of choice NERFINISHED ⓘ |
| implies | If each of two sets can be injected into the other, then the sets have the same cardinality. ⓘ |
| namedAfter |
Ernst Schröder
NERFINISHED
ⓘ
Felix Bernstein NERFINISHED ⓘ Georg Cantor NERFINISHED ⓘ |
| originallyProvedBy | Felix Bernstein NERFINISHED ⓘ |
| proves | antisymmetry of the injection-based preorder on cardinalities ⓘ |
| relatedTo |
Cantor theorem
NERFINISHED
ⓘ
Zermelo–Fraenkel set theory NERFINISHED ⓘ axiom of choice ⓘ well-ordering theorem NERFINISHED ⓘ |
| statedAs | If there exists an injective function from set A to set B and an injective function from set B to set A, then there exists a bijective function between A and B. ⓘ |
| subject |
bijections
ⓘ
cardinality ⓘ equipotence of sets ⓘ injections ⓘ |
| typeOfResult | equivalence theorem ⓘ |
| usedIn |
functional analysis
ⓘ
general topology ⓘ measure theory ⓘ model theory ⓘ theory of cardinal arithmetic ⓘ |
| usesConcept |
bijective function
ⓘ
cardinal number ⓘ injective function ⓘ partial order on cardinalities ⓘ |
| yearProved | 1897 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.