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	4
Cantor–Bernstein–Schröder theorem canonical	1
Schröder–Bernstein theorem	1

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem ⓘ theorem in set theory ⓘ
alsoKnownAs	Bernstein theorem ⓘ Cantor–Bernstein–Schröder theorem ⓘ surface form: Cantor–Bernstein 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 ⓘ surface form: Cantor's theorem Cantor–Bernstein–Schröder theorem self-linksurface differs ⓘ surface form: Schröder–Bernstein theorem Hausdorff maximal principle ⓘ surface form: Zorn's lemma axiom of choice ⓘ axiom of choice ⓘ surface form: well-ordering theorem
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 ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Georg Cantor → knownFor → Cantor–Bernstein–Schröder theorem ⓘ

Felix Bernstein → notableWork → Cantor–Bernstein–Schröder theorem ⓘ

this entity surface form: Cantor–Bernstein theorem

Felix Bernstein → notableConcept → Cantor–Bernstein–Schröder theorem ⓘ

this entity surface form: Cantor–Bernstein theorem

Bernstein → notableFor → Cantor–Bernstein–Schröder theorem ⓘ

subject surface form: Felix Bernstein

this entity surface form: Cantor–Bernstein theorem

Cantor–Bernstein–Schröder theorem → alsoKnownAs → Cantor–Bernstein–Schröder theorem ⓘ

this entity surface form: Cantor–Bernstein theorem

Cantor–Bernstein–Schröder theorem → relatedTo → Cantor–Bernstein–Schröder theorem self-linksurface differs ⓘ

this entity surface form: Schröder–Bernstein theorem