I'm curious: when you define addition after the Peano axioms, the first defining axiom typically goes something like: x + 1 = Sx. Doesn't this say something about '1', at least in relation to addition? Meaning, once you (a) define subtraction (not as the addition of inverses, since there are no inverses in N; just as the solution for x of the equation y = x + a, whenever there is one), and (b) define a total order relation >=, ... aren't you tempted to define some kind of 'integer metric' d(), and say that 1 = d(x,Sx), thus giving some meaning to '1'? After all, when Peano axioms are extended to include (or start with) 0, the nature of 0 as an identity element (with respect to addition) comes naturally out of the axioms and the definition of addition. I would think that some relation between Sx and 1 would come out as well.