Axiom of countable choice

The axiom of countable choice or axiom of denumerable choice is an axiom of set theory similar to the axiom of choice. It states that a countable collection of sets must have a choice function. Paul Cohen showed that this is not provable in ZF. This axiom is required for the devlopment of analysis; in particular, many results depend on having a choice function for a countable set of real numbers.

The axiom of choice clearly implies the axiom of dependent choice, and the axiom of dependent choice is sufficent to show the axiom of choice. The axiom of countable choice is strictly weaker than each of these axioms.