axiom of choice
E87367
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.
Aliases (3)
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
→
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 → 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
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|
| hasStrongerForm |
global axiom of choice
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|
| hasWeakerForm |
countable axiom of choice
→
dependent choice → |
| implies |
Banach–Tarski paradox
→
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 without choice
→
|
| influences |
foundations of mathematics
→
|
| introducedBy |
Ernst Zermelo
→
|
| logicalType |
existential axiom
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|
| motivatedBy |
well-ordering theorem
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|
| 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 (8)
| Subject (surface form when different) | Predicate |
|---|---|
|
axiom of choice
("well-ordering theorem")
→
axiom of choice ("Zorn's lemma") → |
equivalentTo |
|
Ernst Zermelo
("well-ordering theorem")
→
Ernst Zermelo → |
knownFor |
|
axiom of choice
("Hausdorff maximal principle")
→
|
implies |
|
set theory
("Zorn's lemma")
→
|
includesConcept |
|
Ernst Zermelo
("well-ordering theorem")
→
|
proved |
|
von Neumann paradox in set theory
→
|
usesAxiom |