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 nonempty 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 nonempty set for every n ∈ N, there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.
YouTube Encyclopedic

1/3Views:6 991147 91215 595

Choice Functions & The Axiom of Choice  Nathan Dalaklis

How the Axiom of Choice Gives Sizeless Sets  Infinite Series

1.11.3 Countable Sets: Video
Transcription
Overview
The axiom of countable choice (AC_{ω}) 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 AC_{ω}, is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice (Potter 2004). 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 Dedekindinfinite (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 \ {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 AC_{ω}. For other statements equivalent to AC_{ω}, 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 V_{ω}− {Ø} 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 Dedekindinfinite:
 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 nonempty. 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 11 map from X into X which is not onto, proving that X is Dedekindinfinite.
References
 Jech, Thomas J. (1973). The Axiom of Choice. North Holland. pp. 130–131. ISBN 9780486466248.
 Herrlich, Horst (1997). "Choice principles in elementary topology and analysis" (PDF). Comment.Math.Univ.Carolinae. 38 (3): 545.
 Howard, Paul; Rubin, Jean E. (1998). "Consequences of the axiom of choice". Providence, R.I. American Mathematical Society. ISBN 9780821809778.
 Potter, Michael (2004). Set Theory and its Philosophy : A Critical Introduction. Oxford University Press. p. 164. ISBN 9780191556432.
This article incorporates material from axiom of countable choice on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.