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.
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 |
How this entity was disambiguated
This entity first appeared as the object of triple T739664 — 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: axiom of choice Context triple: [von Neumann paradox in set theory, usesAxiom, axiom of choice]
-
A.
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.
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B.
axiom schema of separation
The axiom schema of separation is a principle in set theory that guarantees the existence of subsets defined by properties or predicates, helping to avoid paradoxes by restricting unrestricted set formation.
-
C.
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.
-
D.
set theory
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
-
E.
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.
- 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: axiom of choice Target entity description: 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.
-
A.
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.
-
B.
axiom schema of separation
The axiom schema of separation is a principle in set theory that guarantees the existence of subsets defined by properties or predicates, helping to avoid paradoxes by restricting unrestricted set formation.
-
C.
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.
-
D.
set theory
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
-
E.
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.
- F. None of above. chosen
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 ⓘ |
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: axiom of choice Description of subject: 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.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
well-ordering theorem
this entity surface form:
well-ordering theorem
this entity surface form:
Zorn's lemma
this entity surface form:
well-ordering theorem
this entity surface form:
Zorn's lemma
this entity surface form:
Hausdorff maximal principle
this entity surface form:
well-ordering theorem
this entity surface form:
axiom of choice (over ZF)