A set A is said to be Dedekind-infinite if there is an injective function f:ωA, where ω 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 infiniteMathworldPlanetmath, 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 ChoiceMathworldPlanetmath, 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