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Full Idea
Each proposed revision of set theory is unnatural, because the natural scheme is the unrestricted one that the antinomies discredit.
Gist of Idea
The set scheme discredited by paradoxes is actually the most natural one
Source
Willard Quine (The Ways of Paradox [1961], p.16)
Book Ref
Quine,Willard: 'Ways of Paradox and other essays' [Harvard 1976], 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.
| 21689 | A barber shaves only those who do not shave themselves. So does he shave himself? [Quine] |
| 21690 | Whenever the pursuer reaches the spot where the pursuer has been, the pursued has moved on [Quine] |
| 21691 | Antinomies contradict accepted ways of reasoning, and demand revisions [Quine] |
| 21692 | If we write it as '"this sentence is false" is false', there is no paradox [Quine] |
| 21693 | Russell's antinomy challenged the idea that any condition can produce a set [Quine] |
| 21694 | Membership conditions which involve membership and non-membership are paradoxical [Quine] |
| 21695 | The set scheme discredited by paradoxes is actually the most natural one [Quine] |