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| 21695 | The set scheme discredited by paradoxes is actually the most natural one |
| 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 |
| 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. |
| 21691 | Antinomies contradict accepted ways of reasoning, and demand revisions |
| Full Idea: An 'antinomy' produces a self-contradiction by accepted ways of reasoning. It establishes that some tacit and trusted pattern of reasoning must be made explicit and henceforward be avoided or revised. | |||
| From: Willard Quine (The Ways of Paradox [1961], p.05) | |||
| A reaction: Quine treats antinomies as of much greater importance than mere paradoxes. It is often possible to give simple explanations of paradoxes, but antinomies go to the root of our belief system. This was presumably Kant's intended meaning. |
| 21690 | Whenever the pursuer reaches the spot where the pursuer has been, the pursued has moved on |
| Full Idea: The Achilles argument is that (if the front runner keeps running) each time the pursuer reaches a spot where the pursuer has been, the pursued has moved a bit beyond. | |||
| From: Willard Quine (The Ways of Paradox [1961], p.03) | |||
| A reaction: Quine is always wonderfully lucid, and this is the clearest simple statement of the paradox. |
| 21689 | A barber shaves only those who do not shave themselves. So does he shave himself? |
| Full Idea: In a certain village there is a barber, who shaves all and only those men in the village who do not shave themselves. So does the barber shave himself? The barber shaves himself if and only if he does not shave himself. | |||
| From: Willard Quine (The Ways of Paradox [1961], p.02) | |||
| A reaction: [Russell himself quoted this version of his paradox, from an unnamed source] Quine treats his as trivial because it only concerns barbers, but the full Russell paradox is a major 'antinomy', because it concerns sets. |
| 21694 | Membership conditions which involve membership and non-membership are paradoxical |
| Full Idea: With Russell's antinomy, ...each tie the trouble comes of taking a membership condition that itself talks in turn of membership and non-membership. | |||
| From: Willard Quine (The Ways of Paradox [1961], p.13) | |||
| A reaction: Hence various stipulations to rule out vicious circles or referring to sets of the 'wrong type' are invoked to cure the problem. The big question is how strong to make the restrictions. |
| 21692 | If we write it as '"this sentence is false" is false', there is no paradox |
| Full Idea: If we supplant the sentence 'this sentence is false' with one saying what it refers to, we get '"this sentence is false" is false'. But then the whole outside sentence attributes falsity no longer to itself but to something else, so there is no paradox. | |||
| From: Willard Quine (The Ways of Paradox [1961], p.07) | |||
| A reaction: Quine is pointing us towards type theory and meta-languages to solve the problem. We now have the Revenge Liar, and the problem has not been fully settled. |