| > 2.1, 1.99, 2.00001 I'm really not satisfied with saying that a machine that gives smaller intervals like that is fully sufficient, on the other hand that's really all we're doing when we specify digits... > They're roughly the real numbers you'd get from doing actual (classical) physical measurements. I'd argue that a series of physical measurements don't give you a number so much as a probability distribution, even classically. > do you think we can have pi I'm not sure. The problem I have with denying pi is it doesn't make much more sense than denying 1. Base pi is a perfectly rational numbering system. It's perfectly possible to introduce a special 'pi' symbol (rather like 'i') and define rules of arithmetic so that things work with both rationals and rational multiples of pi. And everything I said just applies to infinitely many other irrationals as well (e.g. 2^(1/n)) > It seems infeasible to mechanically determine that an arbitrary program you are given is a valid pi-digit-calculator though. Indeed, this is true even if you replace "pi" with "1" though, that's another reason why having a program that calculates the digits isn't sufficient. > Having a number could be having a finite representation of the number that can be mechanically tested for equality (in time polynomial in the sizes of representations) Yes. > In what? You want the nth digit printed by time O(P(n))? It's a vague definition anyways, polynomial in the sum of everything that is relevant... probably O(P(n + the number represented)). |