Patricia Shanahan wrote: > I am very familiar with the difficulties of arithmetic with a bounded > range. You lose all sorts of useful properties, such as associativity > and commutativity of addition. Adding up numbers in one order may lead > to an overflow, adding them in a different order doesn't.
Order doesn't matter if one is working in the natural numbers, of course. In the natural numbers both commutativity and associativity hold, in these forms:
(x)(y)(z)(x + y = z ==> y + x = z) (u)(x)(y)(z)(u + (x + y) = z ==> (u + x) + y = z)