Dedekind-infinite

A set $A$ is said to be Dedekind-infiniteif there is an injective function $f\\colon\\omega\\to A$ ,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 thata set is Dedekind-infinite if and only if it is infinite.

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

单词	Dedekindinfinite
释义	Dedekind-infinite A set $A$ is said to be Dedekind-infiniteif there is an injective function $f\\colon\\omega\\to A$ ,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 thata set is Dedekind-infinite if and only if it is infinite. It is consistent with ZF thatthere is an infinite set that is not Dedekind-infinite.However, the existence of such a set requires the failurenot just of the full Axiom of Choice, but even of the Axiom of Countable Choice.
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