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 |
How this entity was disambiguated
This entity first appeared as the object of triple T6929693 — 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: Bernstein theorem Context triple: [Cantor–Bernstein–Schröder theorem, alsoKnownAs, Bernstein theorem]
-
A.
Bernstein inequalities
Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
-
B.
Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
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C.
Hadamard three-circle theorem
The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
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D.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
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E.
Stone–Weierstrass theorem
The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.
- 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: Bernstein theorem Target entity description: 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.
-
A.
Bernstein inequalities
Bernstein inequalities are fundamental results in approximation theory and probability that provide bounds on the derivatives or deviations of functions and random variables under certain smoothness or moment conditions.
-
B.
Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
-
C.
Hadamard three-circle theorem
The Hadamard three-circle theorem is a result in complex analysis that describes how the maximum modulus of a holomorphic function behaves logarithmically between three concentric circles in the complex plane.
-
D.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
E.
Stone–Weierstrass theorem
The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.
- F. None of above. chosen
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 ⓘ |
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: Bernstein theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.