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gptkbp:instanceOf
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gptkb:Titan
|
|
gptkbp:allows
|
most mathematicians
|
|
gptkbp:category
|
gptkb:logic
foundations of mathematics
|
|
gptkbp:controversy
|
non-constructive nature
|
|
gptkbp:equivalentTo
|
gptkb:Well-ordering_theorem
gptkb:Zorn's_Lemma
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gptkbp:field
|
gptkb:set_theory
|
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gptkbp:firstPublicationTitle
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Beweis, dass jede Menge wohlgeordnet werden kann
|
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gptkbp:firstPublished
|
gptkb:Mathematische_Annalen
1904
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|
gptkbp:formedBy
|
gptkb:Ernst_Zermelo
1904
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
Axiom of Choice (AC)
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|
gptkbp:implies
|
every set can be well-ordered
every product of non-empty sets is non-empty
every surjective function has a right inverse
every vector space has a basis
|
|
gptkbp:independenceDate
|
gptkb:Zermelo-Fraenkel_set_theory_(ZF)
|
|
gptkbp:influenced
|
gptkb:algebra
gptkb:topology
analysis
development of modern set theory
|
|
gptkbp:notation
|
∀A (A ≠ ∅ ⇒ ∃f: A → ⋃A, ∀x ∈ A, f(x) ∈ x)
|
|
gptkbp:opposedBy
|
gptkb:L.E.J._Brouwer
constructivists
|
|
gptkbp:relatedAxiom
|
gptkb:Axiom_of_Global_Choice
gptkb:Axiom_of_Countable_Choice
gptkb:Axiom_of_Dependent_Choice
|
|
gptkbp:relatedConcept
|
gptkb:Axiom_of_Determinacy
gptkb:Axiom_of_Countable_Choice
gptkb:Dependent_Choice
|
|
gptkbp:relatedParadox
|
gptkb:Banach–Tarski_paradox
gptkb:Hausdorff_paradox
|
|
gptkbp:sentence
|
Given any collection of nonempty sets, there exists a choice function selecting one element from each set.
|
|
gptkbp:statusInConstructiveMathematics
|
not accepted
|
|
gptkbp:statusInNBG
|
included
|
|
gptkbp:statusInZF
|
independent
|
|
gptkbp:statusInZFC
|
included
|
|
gptkbp:symbol
|
gptkb:AC
|
|
gptkbp:usedIn
|
proof of Hahn–Banach theorem
proof of every vector space has a basis
proof of Tychonoff's theorem
proof of Tychonoff's theorem in topology
proof of existence of non-measurable sets
|
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gptkbp:bfsParent
|
gptkb:constructible_universe_(L)
|
|
gptkbp:bfsLayer
|
7
|