[peteretep]

What a shame you're getting downvoted, simply because you're wrong.

    > Without the decimal point these real
    > numbers just become natural numbers.

If your argument is true, presumably you could write a simple program that would generate all the real numbers with a single, infinite loop?

I wonder how you'd manage to generate 0.1 and 1.0 with your scheme.

[andrewla]

I agree that the downvoting is unfortunate. I guess the assumption is that the post is simply a troll, which seems to be backed up by some of the down-thread replies. Even so, the original post just seems like a list of common misunderstandings about Cantor's notion of infinity and how it corresponds to the way that we use the word in colloquial use.

In defense of the GP, I think they were simply thinking of numbers in [0,1). If you write them backwards, it almost seems like it would work:

    1    ->  .1
    2    ->  .2
    ...
    9    ->  .9
    10   ->  .01
    11   ->  .11
    12   ->  .21
    ...
    3124 ->  .4213
    ...

Done!

Unfortunately, you are either stuck with the fact that some numbers (even simple rational ones) do not have a finite decimal expansion. In most formal proofs of the diagonal theorem, we use the infinite representation without trailing 0s (using trailing 9's instead) to force uniqueness, which makes it even trickier.

Or you're stuck trying to assume that there are natural numbers with an infinite representation, so the decimal representation of the rational number 1/3 would correspond to an actual natural number.

[jerf]

"What a shame you're getting downvoted, simply because you're wrong."

It's not simply because of being wrong. Being wrong is a hazard to your karma, yes, even sometimes just asking questions can be a hazard (which I dislike and do what little I can to fight, but it's still obviously true). But there's a much bigger hazard being invoked here.

[bllguo]

That is extremely disingenuous of you. He's being downvoted for his massive arrogance.