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.
The Axiom of CountableChoice (CC) is a weak form of the Axiom of Choice.
ZF+CC (that is, the Zermelo-Fraenkel axioms together with the Axiom of CountableChoice) suffices to prove that the union of countably many countable sets is countable.
This is version 11 of axiom of countablechoice, born on 2004-10-25, modified 2006-01-05.
, or axiom of denumerable choice, is an axiom of set theory, similar to the axiom of choice.
AC clearly implies the axiom of dependent choice (DC), and DC is sufficient to show AC However AC is strictly weaker than DC (and DC is strictly weaker than AC).
This article incorporates material from axiom of countablechoice on PlanetMath, which is licensed under the GFDL.
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