ABSTRACT
In an earlier paper (1987) 1 adumbrated a conception of arithmetical truth with respect to which the first-order axiom system of Peano arithmetic was to be seen as complete. Given the existence of sentences in the language of arithmetic which are not decided by Peano arithmetic, it must be, of course, that on this conception being expressible by a sentence in a first-order language of arithmetic (say with nonlogical constants 0, S, +, . , <) does not itself render a proposition arithmetical, though this condition is indeed necessary (that a proposition, to be arithmetical, must be expressible in a first-order language in which quantification is over the natural numbers reflects the minimum condition that an arithmetical proposition must be about the natural numbers).’ The further (inductive) condition for a proposition to be arithmetical is that its truth or falsity be perceivable directly on the basis of an articulation of our grasp of the fundamental nature and structure of the natural numbers, or directly from statements which themselves are arithmetical.
