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Full Idea
Sets, unlike extensions, fail to correspond to all concepts. We can prove in ZFC that there is no set corresponding to the concept 'set' - that is, there is no set of all sets.
Gist of Idea
ZFC can prove that there is no set corresponding to the concept 'set'
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.90
A Reaction
This is rather an important point for Frege. However, all concepts have extensions, but they may be proper classes, rather than precisely defined sets.
| 15510 | Classes are a host of ethereal, platonic, pseudo entities [Goodman] |
| 7785 | The use of plurals doesn't commit us to sets; there do not exist individuals and collections [Boolos] |
| 15508 | If singletons are where their members are, then so are all sets [Lewis] |
| 15514 | A huge part of Reality is only accepted as existing if you have accepted set theory [Lewis] |
| 15523 | Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it [Lewis] |
| 13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
| 9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
| 10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |