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Full Idea
We know a great deal about a class without enumerating its members …so definition by extension is not necessary to knowledge about a class ..but enumeration of infinite classes is impossible for finite beings, so definition must be by intension.
Gist of Idea
We can enumerate finite classes, but an intensional definition is needed for infinite classes
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], II)
Book Ref
Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.13
A Reaction
Presumably mathematical induction (which keeps apply the rule to extend the class) will count as an intension here.
| 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] |