The Julius Caesar Objection

Abstract Recent research has revealed three important points about Frege’s philosophy of arithmetic. First, his attempt to derive axioms for arithmetic from principles of logic does not require Frege to appeal to his Axiom V, the axiom which gives rise to Russell’s Paradox. The proofs sketched in Die Grundlagen der Arithmetik depend only upon what (alluding to Frege’s method of introducing it) may be called Hume’s Principle: The number of Fs is the same as the number of Gs just in case there is a one-one correspondence between the Fs and the Gs Formally, the relevant result is that, if a formalization of this Principle is added as an axiom to standard, axiomatic second-order logic, second-order arithmetic can be interpreted in the resulting theory .Secondly, this theory—which may be called Fregean Arithmetic—is itself interpretable in second-order arithmetic and so, presumably, is consistent.3 And thirdly, Frege’s own formal proofs of axioms for arithmetic, given in his Grundgesetze der Arithmetik, do not depend essentially upon Axiom V.4 Indeed, Frege himself knew that he did not require any more than Hume’s Principle, this being essential if he is to draw certain of the philosophical conclusions he wishes to base upon his formal results.