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Full Idea
The naïve view of set theory (that any zero or more things form a set) is natural, but inconsistent: the things that do not belong to themselves are some things that do not form a set.
Gist of Idea
Naïve sets are inconsistent: there is no set for things that do not belong to themselves
Source
George Boolos (Must We Believe in Set Theory? [1997], p.127)
Book Ref
Boolos,George: 'Logic, Logic and Logic' [Harvard 1999], p.127
A Reaction
As clear a summary of Russell's Paradox as you could ever hope for.
| 15894 | Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine] |
| 21695 | The set scheme discredited by paradoxes is actually the most natural one [Quine] |
| 10485 | Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos] |
| 13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
| 9933 | The paradoxes are only a problem for Frege; Cantor didn't assume every condition determines a set [Burgess/Rosen] |
| 9613 | Naïve set theory assumed that there is a set for every condition [Brown,JR] |
| 9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
| 23445 | Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo] |
| 23623 | Predicativism says only predicated sets exist [Hossack] |