# Dedekind-infinite

A set $A$ is said to be *Dedekind-infinite*
if there is an injective function $f:\omega \to A$,
where $\omega $ denotes the set of natural numbers.
A set that is not Dedekind-infinite is said to be *Dedekind-finite*.

A Dedekind-infinite set is clearly infinite^{},
and in ZFC it can be shown that
a set is Dedekind-infinite if and only if it is infinite.

It is consistent with ZF that
there is an infinite set that is not Dedekind-infinite.
However, the existence of such a set requires the failure
not just of the full Axiom of Choice^{}, but even of the Axiom of Countable Choice.

Title | Dedekind-infinite |
---|---|

Canonical name | Dedekindinfinite |

Date of creation | 2013-03-22 12:05:25 |

Last modified on | 2013-03-22 12:05:25 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 11 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E99 |

Synonym | Dedekind infinite |

Related topic | Cardinality |

Defines | Dedekind-finite |

Defines | Dedekind finite |