Predicative fragments of Frege arithmetic

Bulletin of Symbolic Logic 10 (2):153-174 (2004)
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Abstract

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension.

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Øystein Linnebo
University of Oslo

Citations of this work

Properties and the Interpretation of Second-Order Logic.B. Hale - 2013 - Philosophia Mathematica 21 (2):133-156.
Impredicative Identity Criteria.Leon Horsten - 2010 - Philosophy and Phenomenological Research 80 (2):411-439.

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References found in this work

Frege’s Conception of Numbers as Objects.Crispin Wright - 1983 - Critical Philosophy 1 (1):97.
Nominalist Platonism.George Boolos - 1985 - Philosophical Review 94 (3):327-344.
Parts of Classes.Michael Potter - 1993 - Philosophical Quarterly 43 (172):362-366.
From Frege to Gödel.Jean van Heijenoort - 1968 - Philosophy of Science 35 (1):72-72.

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