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Full Idea
In the early versions of set theory ('naïve' set theory), the axiom of comprehension assumed that for any condition there is a set of objects satisfying that condition (so P(x)↔x∈{x:P(x)}), but this led directly to Russell's Paradox.
Clarification
See Idea 6407 for Russell's Paradox
Gist of Idea
Naïve set theory assumed that there is a set for every condition
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.19
A Reaction
How rarely any philosophers state this problem clearly (as Brown does here). This is incredibly important for our understanding of how we classify the world. I'm tempted to just ignore Russell, and treat sets in a natural and sensible way.
Related Idea
Idea 6407 The class of classes which lack self-membership leads to a contradiction [Russell, by Grayling]
| 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] |