Axiom of countable choice


The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. I.e., given a function A with domain N such that A is a non-empty set for every n ∈ N, then there exists a function f with domain N such that f ∈ A for every n ∈ N.

Overview

The axiom of countable choice is strictly weaker than the axiom of dependent choice, which in turn is weaker than the axiom of choice. Paul Cohen showed that ACω, is not provable in Zermelo–Fraenkel set theory without the axiom of choice. 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.
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 \ , one needs 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 and.
A common misconception is that countable choice has an inductive nature and is therefore provable as a theorem 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, and it is this latter result 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− and the set of proper and bounded open intervals of real numbers with rational endpoints.

Use

As an example of an application of ACω, here is a proof that every infinite set is Dedekind-infinite: