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Dedekind-infinite
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(Definition)
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A set $A$ is said to be Dedekind-infinite if 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 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.
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"Dedekind-infinite" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: cardinality
| Other names: |
Dedekind infinite |
| Also defines: |
Dedekind-finite, Dedekind finite |
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Cross-references: axiom of countable choice, axiom of choice, consistent, ZFC, infinite, natural numbers, injective function
There are 2 references to this entry.
This is version 7 of Dedekind-infinite, born on 2002-01-03, modified 2009-01-27.
Object id is 1182, canonical name is DedekindInfinite.
Accessed 4741 times total.
Classification:
| AMS MSC: | 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous) |
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Pending Errata and Addenda
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