# 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. 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**.

## 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 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, Thomas J. (1973).
*The Axiom of Choice*. North Holland. pp. 130–131. ISBN 978-0-486-46624-8. - 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. ISBN 978-0-8218-0977-8. - Potter, Michael (2004).
*Set Theory and its Philosophy : A Critical Introduction*. Oxford University Press. p. 164. ISBN 9780191556432.

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