A
N
N
A(n)
n\inN
f
N
f(n)\inA(n)
n\inN
x
S\subseteqR
S\setminus\{x\}
The ability to perform analysis using countable choice has led to the inclusion of ACω as an axiom in some forms of constructive mathematics, despite its assertion that a choice function exists without constructing it.
As an example of an application of ACω, here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite:
Let
X
n
An
n
X
X
An
(Bn)n\inN
Bn
n
(bn)n\inN
X
(cn)n\inN
This
i
cn
Bn+1
n
X
cn
cn+1
X
X
X
X
The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC), which in turn is weaker than the axiom of choice (AC). DC, and therefore also ACω, hold in the Solovay model, constructed in 1970 by Robert M. Solovay as a model of set theory without the full axiom of choice, in which all sets of real numbers are measurable.
Urysohn's lemma (UL) and the Tietze extension theorem (TET) are independent of ZF+ACω: there exist models of ZF+ACω in which UL and TET are true, and models in which they are false. Both UL and TET are implied by DC.
Paul Cohen showed that ACω is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice. However, some countably infinite sets of non-empty sets can be proven to have a choice function in ZF without any form of the axiom of choice. For example,
V\omega\setminus\{\emptyset\}
V\omega
ZF+ACω suffices to prove that the union of countably many countable sets is countable. These statements are not equivalent: Cohen's First Model supplies an example where countable unions of countable sets are countable, but where ACω does not hold.
There are many equivalent forms to the axiom of countable choice, in the sense that any one of them can be proven in ZF assuming any other of them. They include the following:
(\aleph0)