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2 ideas
| 21695 | The set scheme discredited by paradoxes is actually the most natural one [Quine] |
| Full Idea: Each proposed revision of set theory is unnatural, because the natural scheme is the unrestricted one that the antinomies discredit. | |
| From: Willard Quine (The Ways of Paradox [1961], p.16) | |
| A reaction: You can either takes this free-far-all version of set theory, and gradually restrain it for each specific problem, or start from scratch and build up in safe steps. The latter is (I think) the 'iterated' approach. |
| 21693 | Russell's antinomy challenged the idea that any condition can produce a set [Quine] |
| 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. | |
| From: Willard Quine (The Ways of Paradox [1961], 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. |