If you're having math problems, I feel bad for you, son...

[napoleon_in_rags]

5. "there do not exist 2 integers, say n and m, such that pi can be written as n/m"

Last edited Sun Mar 25, 2012, 05:59 PM - Edit history (1)

What do you mean? I just gave them! 314159.../100000... Two infinitely long integers that when divided equal pi. But you don't like. Why? I can give a way of counting that will include 314159... (it would count all the finite algorithms that count infinitely long integers) so prove its countable. Yet you know this is wrong, you know Z plus all integers of infinite length would probably have the same cardinality as R. And I don't blame you for not liking that.

Ignore my comments on .999 being part of any set (it can be argued both ways) and look at the notion of countability itself. You right, its about creating a 1-to-1 map between the naturals and some set of numbers. But I say look deeper at that idea of a map. I say you need to be able to specify, in finite time, each number of the new set for each natural. Because if a number can't be specified in finite time, its really questionable as to whether its really been specified at all.

You may be thinking about the number e for instance. It is defined through an infinite series, with the dreaded '...' at the end. But what does the "..." mean? It means "you see the pattern." and because of that, you can write it in sigma notation as an infinite sum, using finite information. You can specify it precisely in finite time, same with pi. However if I told you had a number like e, except that the denominator of each part of the infinite sum was TOTALLY RANDOM, it could not be expressed with finite information, because by definition of randomness, there is no simple pattern which can be reduced down to a finite algorithm.

So that's my point: For x to BE a countable number it has to have a finite specification in the family of numbers it belongs. So 314159... is not a an integer, 314159.../100000... is not a rational and 3.14159... is not a decimal. The "..." is a social agreement, saying "you see the pattern" but neither of these numbers really says for sure what the pattern is. The form of e here in sigma notation IS precisely specified with finite information:
http://mathworld.wolfram.com/e.html
with no "..." at the end. And you can produce the same kind of precise, finite definition for pi, and you can produce it for 0.9999 but as soon as you do that, you are giving more information than you had with the "..." hand wavy notation.

The issue isn't about whether 0.999... is equal to 1, the issue is that 0.999... was never a precisely specified number in the first place, anymore than 0.134250978... (you know the rest) is. So the question is, what finite form are you going to reduce it to for me? If you assumed 1/3 = 0.333... and say therefore 0.999... = 3*1/3, of course its equal to 1. If you are defining it by being infinitesimally off from 1, than its infinitesimally off from 1.