math is a game... play around with it, but make sure to follow the rules
I grapple with the absurdity of a post that first expounds mathematical rigor, then does some proof by hand-waving. Pretending that you can use the mathematics of limits without being tied down by the concept that you may not actually be able to evaluate the limit itself is fallacy, and you certainly can't assume it evaluates to it just because you can show that it's a limit.
Indeed the whole concept is more than a little flawed, because when we talk about real numbers, we're implicitly discussing the Real field <R,+,*>. You can't use these operations on infinity without first defining them, and then ensuring that you maintain that every non-zero element in R has a multiplicative inverse where both operations are also commutative. (we need left and right distributivity as well to maintain that R is a field in order that some normal mathematics keeps working on it).
In particular, you're going to run headlong into a huge great headache with maintaining your multiplicative inverse for non-zero elements. You're mathematically required to have an element such that infinity*x = 1, but by the previous equation, 1/infinity = 0. unfortunately this *completely* breaks the requirement for a ring that for any element a, a*0 = 0.
While I'm at it, I might as well mention Cantor's countability proof. N (natural numbers), Z (integers) and Q (rational numbers) are countably infinite. R is not countably infinite, so it's not even correct to assert that there are as many Real numbers as there are rational numbers, because there are infinitely more.
But just for fun I'll throw a wrench in the works by giving you a random way of picking natural numbers (which should give you methods of picking integers and rationals). Start with 1. roll a die, if the result > 3, add 1 and repeat, otherwise stop (that's not the definition of random from statistics, I know, but result isn't deterministic and is directly based off the flat probability of the different results of the dice roll). It'll give you a poisson (?) distribution for the probability of picking any one number, but does theoretically give you the opertunity to randomly choose a number in an infinite set (not with even probability I'll grant you). Unfortunately, I doubt anyone could repeat this for the real number line, since it's not countable and therefore probably couldn't be approached algorithmically anyway. If we restrict ourselves to flat probability distributions though, I doubt it would be at all possible to randomly pick from any infinite set, not least because once you have infinitessimal probabilities of any element occuring, the definition of random goes out the window because rather than being defined as being the same, all the probabilities are defined as something that is undefined.
Anyway. the point of this whole braintwisting nastyness is surely the concept that 1) any particular real number is one of a set that is much larger than those of the rationals, 2) if one were able to assign elements of rational or real number sets to anyone randomly, the chances that anyone else would have the same number as you would be infinitesimal, and given the finite nature of humanity would imply that duplication would never happen in either case, therefore to the finite perception of man (or woman?) be indistinguisable except to those that sit around thinking too much?
Course. I'm just gibbering now.This comment was edited on Feb 24, 21:57.