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Ideas for 'Metaphysics: the logical approach', 'Intro to Gdel's Theorems' and 'Intensional Logic'

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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

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Second-order arithmetic can prove new sentences of first-order [Smith,P]

5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics

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Classical logic is deliberately extensional, in order to model mathematics [Fitting]