nLab
countable choice

The axiom of countable choice, also called ${\mathrm{AC}}_{\omega }$ or ${\mathrm{AC}}_N$, 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.)

More explicitly, let $X$ be any set and let $p:X\to N$ be a surjection. Then the axiom of countable choice states that $p$ has a section.

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?.

category: foundational axiom