Ideas from 'The Philosophy of Mathematics' by Michael Dummett [1998], by Theme Structure
[found in 'Philosophy 2: further through the subject' (ed/tr Grayling,A.C.) [OUP 1998,0-19-875178-8]].
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality
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9194
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The main alternative to ZF is one which includes looser classes as well as sets
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5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
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Intuitionists reject excluded middle, not for a third value, but for possibility of proof
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5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
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First-order logic concerns objects; second-order adds properties, kinds, relations and functions
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5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
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Logical truths and inference are characterized either syntactically or semantically
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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Ordinals seem more basic than cardinals, since we count objects in sequence
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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The number 4 has different positions in the naturals and the wholes, with the same structure
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