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Ideas for 'Parmenides', 'Against Barbaric physics' and 'works'

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4 ideas

5. Theory of Logic / L. Paradox / 3. Antinomies
13986 Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle]
     Full Idea: Plato (in 'Parmenides') shows that the theory that 'Eide' are substances, and Kant that space and time are substances, and Bradley that relations are substances, all lead to aninomies.
     From: report of Plato (Parmenides [c.364 BCE]) by Gilbert Ryle - Are there propositions? 'Objections'
14150 Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato]
     Full Idea: Plato's 'Parmenides' is perhaps the best collection of antinomies ever made.
     From: comment on Plato (Parmenides [c.364 BCE]) by Bertrand Russell - The Principles of Mathematics §337
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox
13483 Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
     Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe).
     From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3
     A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does.
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
8710 The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
     Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself.
     From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics
     A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly?