Predicative Fragments of Frege Arithmetic Øystein Linnebo September 15, 2003 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. 1 Introduction Frege is a logicist about arithmetic: He holds that pure logic provides a possible source of knowledge of arithmetic and that arithmetic therefore is a priori. His defense of this view proceeds in two steps. 1 First he argues that numbers are ascribed to concepts and that the fundamental law governing such ascriptions is Hume’s Principle, 2 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. This principle can be formalized as (HP) Nx.Fx = N x.Gx ↔ F ≈ G where ‘N ’ is an operator that applies to concept variables to form singular terms, and where F ≈ G is a second-order formula saying that there is a relation R that one-to-one correlates the F s and the Gs. 1 Frege gives an informal exposition of his view in [18]; for a more formal treatment, see [19]. 2 Although Cantor’s Principle would have been more accurate historically. 1