April 8
Daniel Isaacson,
Oxford University
Consideration of Dummett's claim that the meaning of 'natural number' is inherently vague
I shall expound Michael Dummett's claim in his paper, 'The philosophical significance of Gödel's theorem' (1963), and in later writings, that a consequence of the indefinite extensibility of Gödel incompleteness is that 'the meaning of 'natural number' is inherently vague'. Though of course Gödel incompleteness establishes that every formal system containing basic arithmetic has a proper extension, I claim, against Dummett's view, that there is a notion of arithmetical truth intrinsic to the meaning of 'natural number' which is stable, not indefinitely extensible, and that first-order Peano Arithmetic is sound and complete with respect to that notion of arithmetical truth. Thereby the meaning of 'natural number' is not vague but clear and precise.