> > I wonder how Nam defines * without using either + or S.
It is undefined in the theory that Shoenfield calls N, which is Robinson's Q.
> Come to think of it, if I can use S, why can't I just > go ahead and define + in the usual (so far as I know) fashion > x + 0 = x > > x + Sy = S(x + y) > > It seems to me that, if one is going to do arithmetic > without addition, unique prime factorization becomes central. > It might be useful to represent numbers by their prime exponents, > so that > 3 * 4 = 12 > becomes > ( 0, 1, ...) * ( 2, 0, ...) = ( 2, 1, ...) > with special rules for 0, of course. It looks like countably > many copies of N, with only finite many copies non-zero. > Each copy has its own successor function S2(x) = 2*x, > S3(x) = 3*x, ... > > However, I am daunted by the prospect of defining * in > this system. We would need to give rules that explain why > 3 + 4 = 7 > or, rather, why > ( 0, 1, 0, 0, ...) + ( 2, 0, 0, 0, ...) = ( 0, 0, 0, 1, ...) > > >
-- Sorrow in all lands, and grievous omens. Great anger in the dragon of the hills, And silent now the earth's green oracles That will not speak again of innocence. David Sutton -- Geomancies