Axiom of countable choice

From Academic Kids

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 development of analysis; in particular, many results depend on having a choice function for a countable set of real numbers (considered as sets of Cauchy sequences of rationals).

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

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