The axiom of countable choice (), also called or , 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).
More explicitly, let be any set and let be a surjection. Then the axiom of countable choice states that has a section. (Recall that a section is a right inverse). In fact, there may be many such sections: having the freedom to choose among them is why this is called “choice”.
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 , is the axiom of dependent choice ().
Sometimes in foundations it is useful to consider a weaker version of countable choice, called . This states that any entire relation from to itself contains a functional entire relation. In terms of surjections, this states that any surjection has a section if is a subset of and is the restriction to of a product projection.
The axiom of weak countable choice () states that a surjection has a section if, whenever , at least one of the preimages and is a singleton. follows (for different reasons) from either or excluded middle. On the other hand, is enough to prove that every Dedekind real number is a Cauchy real number (the converse is always true); more generally, is enough to justify sequential reasoning in analysis. See Bridges et al (1998).