The axiom of countable choice ($CC$), also called $AC_\omega$ or $AC_N$, is a weak form of the axiom of choice; it says that the set of natural numbers is a projective object in Set. (Recall that the full axiom of choice states that every set is projective.)

Definition

More explicitly, let $X$ be any set and let $p\colon X \to \mathbf{N}$ be a surjection. Then the axiom of countable choice states that $p$ has a section. If you phrase the axiom of choice in terms of entire relations, then countable choice states that any entire relation from $\mathbf{N}$ to any other set contains (in the 2-posetRel) a functional entire relation.

Variations

$COSHEP$ & $DC$

Unlike the full axiom of choice, countable choice is often considered to be a constructively acceptable principle. In particular, it does not imply the principle of excluded middle. It is a consequence of COSHEP. A stronger version of countable choice, also a consequence of $COSHEP$, is the axiom of dependent choice ($DC$).

$AC_{00}$

Sometimes in foundations it is useful to consider a weaker version of countable choice, called $AC_{00}$. This states that any entire relation from $\mathbf{N}$ to itself contains a functional entire relation. In terms of surjections, this states that any surjection $p\colon X \to \mathbf{N}$ has a section if $X$ is a subset of $\mathbf{N} \times \mathbf{N}$ and $p$ is the restriction to $X$ of a product projection.

Weak countable choice

The axiom of weak countable choice ($WCC$) states that a surjection $p\colon X \to \mathbf{N}$ has a section if, whenever $m \neq n$, at least one of the preimages$p^*(m)$ and $p^*(n)$ is a singleton. $WCC$ follows (for different reasons) from either $CC$ or excluded middle. On the other hand, $WCC$ is enough to prove that every Dedekind real number is a Cauchy real number (the converse is always true); more generally, $WCC$ is enough to justify sequential reasoning in analysis. See Bridges et al (1998).

References

Douglas Bridges, Fred Richman, and Peter Schuster (1998). A weak countable choice principle. Available from Fred Richman’s Documents.