axiom of countable choice

The axiom of countable choice or axiom of denumerable choice, denoted AC_ω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, 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, there exists a function ''f'' with domain N such that ''f''(''n'') ∈ ''A''(''n'') for every ''n'' ∈ N.

Overview

The axiom of countable choice (AC_ω) is strictly weaker than the axiom of dependent choice (DC), which in turn is weaker than the axiom of choice (AC). Paul Cohen showed that AC_ω is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice . AC_ω holds in the Solovay model.

ZF+AC_ω 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).

AC_ω 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'' \ , 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 AC_ω. For other statements equivalent to AC_ω, see and .

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 non-empty sets can be proven to have a choice function in ZF without ''any'' form of the axiom of choice. These include ''V''_''ω''−  and the set of proper and bounded open intervals of real numbers with rational endpoints.

Use

As an example of an application of AC_ω, here is a proof (from ZF + AC_ω) that every infinite set is Dedekind-infinite:
:Let ''X'' be infinite. For each natural number ''n'', let ''A''_''n'' be the set of all 2^''n''-element subsets of ''X''. Since ''X'' is infinite, each ''A''_''n'' is non-empty. The first application of AC_ω yields a sequence (''B''_''n'' : ''n'' = 0,1,2,3,...) where each ''B''_''n'' is a subset of ''X'' with 2^''n'' elements.
:The sets ''B''_''n'' are not necessarily disjoint, but we can define
:: ''C''₀ = ''B''₀
::''C''_''n'' = the difference between ''B''_''n'' and the union of all ''C''_''j'', ''j'' < ''n''.
:Clearly each set ''C''_''n'' has at least 1 and at most 2^''n'' elements, and the sets ''C''_''n'' are pairwise disjoint. The second application of AC_ω yields a sequence (''c''_''n'': ''n'' = 0,1,2,...) with c_''n'' ∈ ''C''_''n''.
:So all the c_''n'' are distinct, and ''X'' contains a countable set. The function that maps each ''c''_''n'' to ''c''_''n''+1 (and leaves all other elements of ''X'' fixed) is a 1-1 map from ''X'' into ''X'' which is not onto, proving that ''X'' is Dedekind-infinite.

References

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{{DEFAULTSORT:Axiom Of Countable Choice
Axiom of choice