axiom of choice

E87367

mathematical principle set-theoretic axiom

The axiom of choice is a fundamental principle in set theory asserting that one can select an element from each set in any collection of nonempty sets, with far-reaching consequences across mathematics.

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All labels observed (5)

Label	Occurrences
axiom of choice canonical	5
well-ordering theorem	4
Zorn's lemma	2
Hausdorff maximal principle	1
axiom of choice (over ZF)	1

Statements (50)

Predicate	Object
instanceOf	mathematical principle ⓘ set-theoretic axiom ⓘ
abbreviationInZFC	the C in ZFC ⓘ
acceptedIn	most mainstream mathematics ⓘ
alsoKnownAs	AC ⓘ
appliesTo	arbitrary collections of nonempty sets ⓘ
centralTo	development of modern set theory ⓘ
consistencyRelativeTo	Zermelo–Fraenkel set theory if ZF is consistent ⓘ
controversialBecause	implies counterintuitive results like Banach–Tarski paradox ⓘ leads to non-constructive existence proofs ⓘ
domainOfDiscourse	collections of nonempty sets ⓘ
equivalentTo	Tychonoff theorem for products of compact spaces ⓘ axiom of choice self-linksurface differs ⓘ surface form: Zorn's lemma every set can be written as a disjoint union of choice sets for a partition ⓘ every surjective function has a right inverse ⓘ every vector space has a basis ⓘ axiom of choice self-linksurface differs ⓘ surface form: well-ordering theorem
expressedAs	every family of nonempty sets admits a choice function ⓘ
field	set theory ⓘ
formalizes	ability to choose an element from each set in a family of nonempty sets ⓘ
hasStrongerForm	global axiom of choice ⓘ
hasWeakerForm	countable axiom of choice ⓘ dependent choice ⓘ
implies	Banach–Tarski paradox ⓘ axiom of choice self-linksurface differs ⓘ surface form: Hausdorff maximal principle every field has an algebraic closure ⓘ every infinite set has a countable subset ⓘ every product of nonempty sets is nonempty ⓘ every set can be well-ordered ⓘ every vector space has a Hamel basis ⓘ existence of non-measurable sets of real numbers ⓘ well-ordering of every set ⓘ
independenceProvedBy	Kurt Gödel ⓘ Paul Cohen ⓘ
independentOf	Zermelo–Fraenkel set theory ⓘ surface form: Zermelo–Fraenkel set theory without choice
influences	foundations of mathematics ⓘ
introducedBy	Ernst Zermelo ⓘ
logicalType	existential axiom ⓘ
motivatedBy	well-ordering theorem ⓘ
quantifierForm	for every family F of nonempty sets there exists a function f with domain F such that f(X) is in X for all X in F ⓘ
rejectedIn	constructive mathematics ⓘ some schools of intuitionism ⓘ
relatedTo	continuum hypothesis ⓘ
statusInZFC	axiom of Zermelo–Fraenkel set theory with choice ⓘ
usedIn	algebra ⓘ category theory ⓘ functional analysis ⓘ measure theory ⓘ topology ⓘ
yearIntroduced	1904 ⓘ

Referenced by (13)

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

von Neumann paradox in set theory → usesAxiom → axiom of choice ⓘ

Ernst Zermelo → knownFor → axiom of choice ⓘ

this entity surface form: well-ordering theorem

Ernst Zermelo → knownFor → axiom of choice ⓘ

Ernst Zermelo → proved → axiom of choice ⓘ

this entity surface form: well-ordering theorem

set theory → includesConcept → axiom of choice ⓘ

this entity surface form: Zorn's lemma

axiom of choice → equivalentTo → axiom of choice self-linksurface differs ⓘ

this entity surface form: well-ordering theorem

axiom of choice → equivalentTo → axiom of choice self-linksurface differs ⓘ

this entity surface form: Zorn's lemma

axiom of choice → implies → axiom of choice self-linksurface differs ⓘ

this entity surface form: Hausdorff maximal principle

Cantor–Bernstein–Schröder theorem → doesNotRequire → axiom of choice ⓘ

Cantor–Bernstein–Schröder theorem → relatedTo → axiom of choice ⓘ

this entity surface form: well-ordering theorem

Cauchy functional equation → pathologicalSolutionsDependOn → axiom of choice ⓘ

Tychonoff theorem for products of compact spaces → equivalentTo → axiom of choice ⓘ

this entity surface form: axiom of choice (over ZF)

Banach–Tarski paradox → assumes → axiom of choice ⓘ