Frege's Natural Numbers: Motivations and Modifications
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Frege on Numbers: Beyond the Platonist Picture*
Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly o...
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By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map ν from the power set of E to E satisfying the condition ∀XY[ ν(X) = ν(Y) ⇔ X ≈ Y] , then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this n...
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Two logics are implicationally equivalent if the axioms and inference rules of each imply the axioms of the other. Characterizing the inferential equivalences of various formulations of the sentential calculi is foundational. Using an automated deduction system, I show that Łukasiewicz's CN can be derived from Frege's Begriffsschrift, the first sentential calculus ; the proof appears to be novel.
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We now know of a number of ways of developing Real Analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in ...
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