Are the "flipped" 10-adic number equivalent to the real segment $[0;1]$

Question

I just came with an question/idea trying to really feel why real numbers a uncountably infinte. I know Cantor's diagonal argument and I'm quite convinced by it, but here is the idea I had which has to be wrong somewhere, but I can't see where.

Let's write $\overline{n}$ the flipped version of some natural number $n$ (so $\overline{6320} = 0236$), and define the function $f$ such that

$f(n) = 0.\overline{n}$

I have the feeling that every real number between $0$ and $1$ will eventually be described if we let $n$ going through all the natural numbers, but then it would create a mapping between $\mathbb{N}$ and the real segment $[0;1]$, and so $\mathbb{R}$...

I think that a way of putting my question would be : are the "flipped" 10-adic number equivalent to the real segment $[0;1]$ ? And if so, why are the reals uncountable ??

Answer 1

To get a intuitive feeling that the reals are uncountably infinite, you need to understand three things:

(1) The power set of any set is strictly larger than the original set. In a sense, this "intuitive feeling" is basically "all possible combination of things in the set" is kind of a dimension higher than the things in the set.

(2) The set of real numbers is the power set of the set of integers, by definition of how the real number works in decimal representation.

(3) The term "uncountably infinite" is just a fancy way to say "strictly larger than the integer set".

Now p-adic number has nothing to do with countability, it has some technical insights that you can infer from the comment of this thread.

Edit: To answer your original question, by definition of how the series representation of the p-adic number is constructed, the set of base-p p-adic numbers is indeed "uncountably infinite", but that's not what they are useful for.