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 Originally Posted by oskar
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.
I get this, but I don't think it has any value whatsoever, and maybe isn't even right, or perhaps the problem is that the mathematical language we have created is finite. It's basically a chicken and an egg thing. Because for any real number that you can come up with that is between two natural numbers I can just add another natural number.
I think it becomes a problem with a definition of infinity. I don't think any infinity can be greater than any other infinty, because doesn't infinity mean no limit? In which case there is no difference between the two. Trying to compare two infinitys is stupid because they are exactly the same thing, quite simply unlimited, so one can't possibly be larger than the other because they aren't a size, they are an absence of one.
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