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Full Idea
A set is 'Dedekind-infinite' iff there exists a one-to-one function that maps a set into a proper subset of itself.
Gist of Idea
An infinite set maps into its own proper subset
Source
report of Richard Dedekind (Nature and Meaning of Numbers [1888], §64) by E Reck / M Price - Structures and Structuralism in Phil of Maths n 7
A Reaction
Sounds as if it is only infinite if it is contradictory, or doesn't know how big it is!
| 10183 | An infinite set maps into its own proper subset [Dedekind, by Reck/Price] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 14427 | We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell] |
| 9944 | We understand some statements about all sets [Putnam] |
| 9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |