The development of arithmetic in Frege's Grundgesetze der arithmetik

Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system-Axiom V-which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, "Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?" ?0. Opening. In his Grundlagen der Arithmetik, Frege explicitly defines numbers as the extensions of concepts of a certain form. The number of Fs is, according to Frege, the extension of the (second-level) concept "is a concept which can be correlated one-one with the concept Fo" (GI, ?68).' Frege then derives, from his explicit definition, what is known as Hume's Principle (see GI, ?73): The number of Fs is the same as the number of Gs just in case the Fs can be correlated one-one with the Gs. Once the proof of Hume's Principle is complete, Frege sketches the proofs of a variety of facts about numbers. This sketch is intended to show that the fundamental laws of arithmetic can be formally derived from his explicit definition. In Received June 12, 1992; revised September 2, 1992. 1 GOTTLOB FREGE, The foundations of arithmetic (J. L. Austin, translator), Northwestern University Press, Evanston, Illinois, 1980. References in the text are marked by GI and a section number. ? 1993, Association for Symbolic Logic 0022-4812/93/5804-0010/$03.30