Quote Originally Posted by kiwiMark View Post
Okay so I'm dumb and he's smart and probably right and all the rest of it but I learn best by assuming that I am king of the world and arguing. As such:

He seems to be saying:

let's take allllllll the real numbers between 0 and 1. (0.4, 0.7235762355 etc.)
let's take allllllll the natural numbers (1, 2, 3 etc.)

So far these two sets are equal. If you like, imagine that you're counting the first set, this automatically generates the second. He's saying if we were to list these out to infinity (also seems troubling that he keeps saying that like it's a logical possibility) then we get all of them.

Then he says, now imagine a real number X that isn't equal to any of the real numbers "already listed".

So now we're rocking it like this:

1. We've already "listed" "all" the natural numbers.
2. Before this whole X thing, we have a "list" of real numbers that is exactly as long as the "list" of "all" the natural numbers.
3. So once we add X to that second list, the second list becomes longer than the first.

This is all well and good but he fails to prove that X can exist.

To my mind, any possible value that X could be, we've already covered since we kept listing real numbers out to infinity.

If you want to say that there's a magical real number X that's not equal to any of the "other" real numbers, why not a magical natural number Y, that's not equal to any of the other natural numbers and restores the 1:1 equilibrium?
I'll just come back to that because it's super easy, and I can feel smart by answering it.
If you want to start representing natural numbers by real numbers, you can just take a natural number that is

2164984351684684684....
where each digit is
1a2a3a4a5a6a
And represent it by a real number that is
0.1a2a3a4a5a6a...
or
0.0001a2a3a4a5a6a...
or
99.123bacon1a2a3a4a5a6a...

You can already see that intuitively you can fit infinite infinites of natural numbers in a real number any which way you like, so proving that there are infiniely more numbers in any arbitrary interval of real numbers than there are in all of N is just a formality.