Dedekind-infinite Set
In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite.
A vaguely related notion is that of a Dedekind-finite ring. A ring is said to be a Dedekind-finite ring if ab=1 implies ba=1 for any two ring elements a and b. These rings have also been called directly finite rings.
Read more about Dedekind-infinite Set: Comparison With The Usual Definition of Infinite Set, Dedekind-infinite Sets in ZF, History, Relation To The Axiom of Choice, Proof of Equivalence To Infinity, Assuming Axiom of Countable Choice, Generalizations
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