Good thread, I love this stuff.
I like the way that Penrose describes it though, it's a bit clearer.
Assume that you've established some one to one corrospondence between the naturals (1, 2, 3...) and the reals. It doesn't really matter what; for example:
1 : .7683...
2 : .7182...
3 : .0100...
4 : .1415...
And we assume that every real number has been included in the list.
However, if we construct a new real by taking the first digit from the first real and the second digit from the second real and so on, we will generate the following real:
.7101...
Now replace every digit 0 in this real by 1 and every digit not 0 by the digit 0:
.0010...
and hence we know by definition that this real is not in the original list, and hence there is no comprehensive mapping between the naturals and the reals.
To go into a little more detail, this is what mathematicians call 'uncountability', because if you think about it, when you count stuff, you are doing exactly this; assigning a natural number to each object. So what we are saying above is that not only are there an infinite number of reals, but you can't even count them, unlike the naturals.
Incedentally the rationals are countable. Say you have the rational
28/135.
Split this into its prime factors (none of which will be shared because they will cancel):
2.2.7 / 3.3.3.5 = 2^2.7 / 3^3.5
Now, double the power of each integer on top, and double and add one of each integer on the bottom. Then multiply them together:
2^4.7^2.3^7.5^3.
Hence you have a unique natural number for every rational number. You can get back to the rational by collecting all even powered numbers, halving the power, and then multiply them together to get the top of the fraction, and collecting all odd powered numbers, subtracting 1 and halving the power, then multiply them together, to get the bottom of the fraction.
Cool stuffs huh?
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