Axiom of countable choice


Axiom of countable choice

The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory, similar to the axiom of choice. It states that any countable collection of non-empty sets must have a choice function. Spelled out, this means that if A is a function with domain N (where N denotes the set of natural numbers) and 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.

Paul Cohen showed that ACω is not provable in Zermelo-Fraenkel set theory without the axiom of choice (ZF).

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 (for arbitrary n) finite choice for a finite set of size n, and it is this latter result (which is an elementary theorem in combinatorics) that is provable by induction.

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 of a set SR is the limit of some sequence of elements of S\{x}, one uses (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)[1] and (Howard/Rubin 1998).[2]

ACω is a weak form of the axiom of choice (AC). AC states that every collection of non-empty sets must have a choice function. AC clearly implies the axiom of dependent choice (DC), and DC is sufficient to show ACω. However ACω is strictly weaker than DC[3] (and DC is strictly weaker than AC).

Use

As an example of an application of ACω, here is a proof (from ZF+ACω) that every infinite set is Dedekind-infinite:

Let X be infinite. For each natural number n, let An be the set of all 2n-element subsets of X. Since X is infinite, each An is nonempty. A first application of ACω yields a sequence (Bn : n=0,1,2,3,...) where each Bn is a subset of X with 2n elements.
The sets Bn are not necessarily disjoint, but we can define
C0 = B0
Cn= the difference of Bn and the union of all Cj, j<n.
Clearly each set Cn has at least 1 and at most 2n elements, and the sets Cn are pairwise disjoint. A second application of ACω yields a sequence (cn: n=0,1,2,...) with cnCn.
So all the cn are distinct, and X contains a countable set. The function that maps each cn to cn+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

  1. ^ Horst Herrlich, Choice principles in elementary topology and analysis, Comment.Math.Univ.Carolinae 38,3 (1997), pp. 545-545
  2. ^ Paul Howard and Jean E. Rubin. Consequences of the axiom of choice. Providence, R.I.: American Mathematical Society, 1998.
  3. ^ Jech T.J., The Axiom of Choice, North Holland, 1973.

This article incorporates material from axiom of countable choice on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.


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