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June 2004 Predicative fragments of Frege Arithmetic
Øystein Linnebo
Bull. Symbolic Logic 10(2): 153-174 (June 2004). DOI: 10.2178/bsl/1082986260

Abstract

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of Fs is identical to the number of Gs if and only if the Fs 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.

Citation

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Øystein Linnebo. "Predicative fragments of Frege Arithmetic." Bull. Symbolic Logic 10 (2) 153 - 174, June 2004. https://doi.org/10.2178/bsl/1082986260

Information

Published: June 2004
First available in Project Euclid: 26 April 2004

zbMATH: 1068.03051
MathSciNet: MR2062415
Digital Object Identifier: 10.2178/bsl/1082986260

Rights: Copyright © 2004 Association for Symbolic Logic

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Vol.10 • No. 2 • June 2004
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