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Full Idea
'Comprehension' is the assumption that every predicate has an extension. Naïve set theory is the theory whose axioms are extensionality and comprehension, and comprehension is thought to be its naivety.
Gist of Idea
Naïve set theory has trouble with comprehension, the claim that every predicate has an extension
Source
William D. Hart (The Evolution of Logic [2010], 1)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.4
A Reaction
This doesn't, of course, mean that there couldn't be a more modest version of comprehension. The notorious difficulty come with the discovery of self-referring predicates which can't possibly have extensions.
| 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] |