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 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 (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 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\{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 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 nonempty. A 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 of 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. A 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
- Jech, T.J. (1973). The Axiom of Choice. North Holland.
- Herrlich, Horst (1997). "Choice principles in elementary topology and analysis" (PDF). Comment.Math.Univ.Carolinae. 38 (3): 545–545.
- Howard, Paul; Rubin, Jean E. (1998). "Consequences of the axiom of choice". Providence, R.I. American Mathematical Society.
- 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/Share-Alike License.