Cantor–Bernstein–Schröder theorem

E160401

mathematical theorem theorem theorem in set theory

The Cantor–Bernstein–Schröder 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.

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All labels observed (3)

Label	Occurrences
Cantor–Bernstein theorem	5
Cantor–Bernstein–Schröder theorem canonical	2
Schröder–Bernstein theorem	2

How this entity was disambiguated

This entity first appeared as the object of triple T1396006 — 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: Cantor–Bernstein–Schröder theorem
Context triple: [Georg Cantor, knownFor, Cantor–Bernstein–Schröder theorem]

A. Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
B. Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
C. Burali-Forti paradox
The Burali-Forti paradox is a foundational logical contradiction in set theory that arises from considering the set of all ordinal numbers, showing that such a totality cannot consistently exist as a set.
D. Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
E. Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
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: Cantor–Bernstein–Schröder theorem
Target entity description: The Cantor–Bernstein–Schröder 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. Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
B. Cantor’s paradox
Cantor’s paradox is a foundational result in set theory showing that the “set of all sets” cannot exist because its power set would have a strictly larger cardinality, leading to a contradiction.
C. Burali-Forti paradox
The Burali-Forti paradox is a foundational logical contradiction in set theory that arises from considering the set of all ordinal numbers, showing that such a totality cannot consistently exist as a set.
D. Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
E. Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem ⓘ theorem in set theory ⓘ
alsoKnownAs	Bernstein theorem ⓘ Cantor–Bernstein theorem ⓘ linked to: Cantor–Bernstein–Schröder theorem
appliesTo	arbitrary sets ⓘ finite sets ⓘ infinite sets ⓘ
classification	result about equivalence relations on sets via bijections ⓘ
concerns	bijective functions ⓘ cardinality of sets ⓘ equipotent sets ⓘ injective functions ⓘ
doesNotRequire	axiom of choice ⓘ
ensuresExistenceOf	bijection between two mutually embeddable sets ⓘ
field	mathematical logic ⓘ set theory ⓘ
formalizes	equivalence of mutual embeddability and equipotence for sets ⓘ
generalFormulation	For sets A and B, if there exists an injection f:A→B and an injection g:B→A, then there exists a bijection h:A↔B. ⓘ
hasAlternativeProofMethod	category-theoretic arguments ⓘ order-theoretic arguments ⓘ
hasStandardProofMethod	construction via chains of elements under injections ⓘ
implies	If each of two sets can be injected into the other, then the two sets have the same cardinality. ⓘ
isFundamentalResultIn	set theory ⓘ
logicalStrength	provable in ZF set theory ⓘ
mathematicalDomain	theory of cardinal numbers ⓘ
namedAfter	Ernst Schröder ⓘ Felix Bernstein ⓘ Georg Cantor ⓘ
originalProofBy	Felix Bernstein ⓘ
relatedTo	Cantor's theorem ⓘ linked to: Cantor’s theorem Schröder–Bernstein theorem ⓘ linked to: Cantor–Bernstein–Schröder theorem Zorn's lemma ⓘ linked to: Hausdorff maximal principle axiom of choice ⓘ well-ordering theorem ⓘ linked to: axiom of choice
statement	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. ⓘ
topic	comparison of cardinalities ⓘ equivalence of sets ⓘ partial order of cardinal numbers ⓘ
usedIn	construction of bijections between infinite sets ⓘ foundations of measure theory ⓘ functional analysis ⓘ proofs about cardinal arithmetic ⓘ theory of equivalence relations on sets ⓘ
usedToShow	mutual injections imply equal cardinality ⓘ
yearFirstPublished	1897 ⓘ

How these facts were elicited