# axiom of countable choice

The *Axiom of Countable Choice* (CC) is a weak form of the Axiom of Choice^{} (http://planetmath.org/AxiomOfChoice).
It states that every countable set of nonempty sets has a choice function.

(that is, the Zermelo-Fraenkel axioms^{} together with the Axiom of Countable Choice) suffices to prove that the union of countably many countable sets is countable^{}. It also suffices to prove that every infinite set^{} has a countably infinite^{} subset, and that a set $X$ is infinite if and only if there is a bijection between $X$ and a proper subset^{} of $X$.

Title | axiom of countable choice |
---|---|

Canonical name | AxiomOfCountableChoice |

Date of creation | 2013-03-22 14:46:23 |

Last modified on | 2013-03-22 14:46:23 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 14 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E25 |

Synonym | countable axiom of choice |

Synonym | countable AC |

Defines | countable choice |