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Full Idea
In the case of Russell's antinomy, the tacit and trusted pattern of reasoning that is found wanting is this: for any condition you can formulate, there is a class whose members are the things meeting the condition.
Gist of Idea
Russell's antinomy challenged the idea that any condition can produce a set
Source
Willard Quine (The Ways of Paradox [1961], p.11)
Book Ref
Quine,Willard: 'Ways of Paradox and other essays' [Harvard 1976], p.11
A Reaction
This is why Russell's Paradox is so important for set theory, which in turn makes it important for the foundations of mathematics.
| 8625 | What physical facts could underlie 0 or 1, or very large numbers? [Frege on Mill] |
| 8469 | Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein] |
| 21693 | Russell's antinomy challenged the idea that any condition can produce a set [Quine] |
| 9406 | A class is natural when everybody can spot further members of it [Quinton] |
| 9984 | We can have a series with identical members [Tait] |
| 17837 | Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M] |
| 8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
| 17824 | The master science is physical objects divided into sets [Maddy] |
| 9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
| 9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |