Dedekind-infinite
A set is said to be Dedekind-infinite if there is an injective function , where 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 |