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- Axiom_of_countable_choice abstract "The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that any countable collection of non-empty sets must have a choice function. I.e., given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, then there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC), (Jech 1973) which in turn is weaker than the axiom of choice (AC). Paul Cohen showed that ACω, is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice(Potter 2004). ACω holds in the Solovay model.ZF + ACω suffices to prove that the union of countably many countable sets is countable. It also suffices to prove that every infinite set is Dedekind-infinite (equivalently: has a countably infinite subset).ACω is particularly useful for the development of analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. For instance, in order to prove that every accumulation point x of a set S⊆R is the limit of some sequence of elements of S\\{x}, one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω. For other statements equivalent to ACω, see Herrlich (1997) and Howard & Rubin (1998).A common misconception is that countable choice has an inductive nature and is therefore provable as a theorem (in ZF, or similar, or even weaker systems) by induction. However, this is not the case; this misconception is the result of confusing countable choice with finite choice for a finite set of size n (for arbitrary n), and it is this latter result (which is an elementary theorem in combinatorics) that is provable by induction. However, some countably infinite sets of nonempty sets can be proven to have a choice function in ZF without any form of the axiom of choice. These include Vω− {Ø} and the set of proper and bounded open intervals of real numbers with rational endpoints.".
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- Axiom_of_countable_choice wikiPageWikiLink Axiom.
- Axiom_of_countable_choice wikiPageWikiLink Axiom_of_choice.
- Axiom_of_countable_choice wikiPageWikiLink Axiom_of_dependent_choice.
- Axiom_of_countable_choice wikiPageWikiLink Category:Axiom_of_choice.
- Axiom_of_countable_choice wikiPageWikiLink Choice_function.
- Axiom_of_countable_choice wikiPageWikiLink Countable_set.
- Axiom_of_countable_choice wikiPageWikiLink Dedekind-infinite_set.
- Axiom_of_countable_choice wikiPageWikiLink Domain_of_a_function.
- Axiom_of_countable_choice wikiPageWikiLink Empty_set.
- Axiom_of_countable_choice wikiPageWikiLink Function_(mathematics).
- Axiom_of_countable_choice wikiPageWikiLink Infinite_set.
- Axiom_of_countable_choice wikiPageWikiLink Interval_(mathematics).
- Axiom_of_countable_choice wikiPageWikiLink Limit_(mathematics).
- Axiom_of_countable_choice wikiPageWikiLink Limit_point.
- Axiom_of_countable_choice wikiPageWikiLink Mathematical_analysis.
- Axiom_of_countable_choice wikiPageWikiLink Metric_space.
- Axiom_of_countable_choice wikiPageWikiLink Natural_number.
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- Axiom_of_countable_choice wikiPageWikiLink Real_number.
- Axiom_of_countable_choice wikiPageWikiLink Sequence.
- Axiom_of_countable_choice wikiPageWikiLink Set_(mathematics).
- Axiom_of_countable_choice wikiPageWikiLink Set_theory.
- Axiom_of_countable_choice wikiPageWikiLink Solovay_model.
- Axiom_of_countable_choice wikiPageWikiLink Zermelo–Fraenkel_set_theory.
- Axiom_of_countable_choice wikiPageWikiLink File:Axiom_of_countable_choice.svg.
- Axiom_of_countable_choice wikiPageWikiLinkText "Axiom of countable choice".
- Axiom_of_countable_choice wikiPageWikiLinkText "Countable Choice".
- Axiom_of_countable_choice wikiPageWikiLinkText "axiom of countable choice".
- Axiom_of_countable_choice wikiPageWikiLinkText "countable choice".
- Axiom_of_countable_choice id "6418".
- Axiom_of_countable_choice title "axiom of countable choice".
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- Axiom_of_countable_choice subject Category:Axiom_of_choice.
- Axiom_of_countable_choice hypernym Axiom.
- Axiom_of_countable_choice type Source.
- Axiom_of_countable_choice comment "The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that any countable collection of non-empty sets must have a choice function.".
- Axiom_of_countable_choice label "Axiom of countable choice".
- Axiom_of_countable_choice sameAs Q1000116.
- Axiom_of_countable_choice sameAs Axiom_spočetného_výběru.
- Axiom_of_countable_choice sameAs Abzählbares_Auswahlaxiom.
- Axiom_of_countable_choice sameAs اصل_انتخاب_شمارا.
- Axiom_of_countable_choice sameAs Assioma_della_scelta_numerabile.
- Axiom_of_countable_choice sameAs 可算選択公理.
- Axiom_of_countable_choice sameAs 가산_선택_공리.
- Axiom_of_countable_choice sameAs m.026gkc.
- Axiom_of_countable_choice sameAs Q1000116.
- Axiom_of_countable_choice sameAs 可数选择公理.
- Axiom_of_countable_choice wasDerivedFrom Axiom_of_countable_choice?oldid=682057667.
- Axiom_of_countable_choice depiction Axiom_of_countable_choice.svg.
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