Predicative fragments of Frege arithmetic

Bulletin of Symbolic Logic 10 (2):153-174 (2004)
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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Richard Heck (2011). Ramified Frege Arithmetic. Journal of Philosophical Logic 40 (6):715-735.
Leon Horsten (2010). Impredicative Identity Criteria. Philosophy and Phenomenological Research 80 (2):411-439.

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