more from this thinker | more from this text
Full Idea
We seem to understand some statements about all sets (e.g. 'for every set x and every set y, there is a set z which is the union of x and y').
Gist of Idea
We understand some statements about all sets
Source
Hilary Putnam (Mathematics without Foundations [1967], p.308)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.308
A Reaction
His example is the Axiom of Choice. Presumably this is why the collection of all sets must be referred to as a 'class', since we can talk about it, but cannot define it.
| 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] |