Russell and Peano on the independence of the axioms of arithmetic
It is usually claimed that the Frege-Russell conception of logic rejects metatheoretical investigations. Although Peano is linked to this conception of logic, he did consider on several occasions the independence of the axioms of geometry and arithmetic. In this talk I shall argue that the general claim that Russell rejects tout court independence proofs should be revised. I shall focus on Russell’s account in The Principles of Mathematics (1903). First, I shall defend that Russell’s structuralist understanding of Peano’s axiomatisation of arithmetic is not fully faithful to Peano’s strategy. Second, I shall explain why Russell does not object to Peano’s strategy of performing independence proofs for the axioms of arithmetic and defend that Russell’s opposition to independence arguments should not be understood in general, but in the context of the axioms of a logical system.