I was recently thinking about the Infinite Hotel Paradox and how it covers countable and uncountable infinities, and I was wondering what's stopping anyone from using the table proof for countable infinity.
Say I rewrite the numbers as such:
...0000000000000000001
...0000000000000000002
...0000000000000000003
...0000000000000000004
.
.
.
...0000000000000000010
...0000000000000000011
...0000000000000000012
.
.
.
These numbers are still exactly the same, albeit with an infinite number of 0's padding them. The number of 0's is countable, the number of numbers is countably infinite, and now watch as I flip them around:
1000000000000000000...
2000000000000000000...
3000000000000000000...
4000000000000000000...
.
.
.
0100000000000000000...
1100000000000000000...
2100000000000000000...
.
.
.
This is still countably infinite no? Now why can't I apply the same table proof that was used to prove that decimals are uncountably infinite to this? Did one of those transformations convert it to uncountable? Because watch what happens when I just:
0.1000000000000000000...
0.2000000000000000000...
0.3000000000000000000...
0.4000000000000000000...
.
.
.
0.0100000000000000000...
0.1100000000000000000...
0.2100000000000000000...
.
.
.
Suddenly we have a way to systematically generate all the decimal numbers by mapping it 1 to 1 with all the whole numbers! Or so I think, I'm definitely wrong on the fact but that's why I'm asking the question: Why doesn't this system work, and at what point did I change it from countable to uncountable?
Now let's think of it a different way. Say I were to group all the whole numbers as so:
[1, 2, 3, 4, 5, 6, 7, 8, 9]
[10, 11, 12, 13, 14, 15, 16...]
[100, 101, 102, 103, 104...]
...
I have grouped them all into how many digits they have, or rather groups of the answer to $\lfloor \log_{10}{x} \rfloor$
Now who says I can't do the same with decimals?
[0.1, 0.2, 0.3, 0.4, 0.5...]
[0.01, 0.02, 0.03, 0.04...]
[0.001, 0.002, 0.003, 0.004...]
...
Instead, I'm counting how many places after there are until the decimal terminates. Would you say that's not also countably infinite?
There's definitely something I'm not getting here, hence why I'm asking where the flaw in my logic is. There's definitely some transformation I'm doing that makes it uncountably infinite but I'm not exactly sure where. Does it maybe have to do with the fact that there's no end goal for countable infinity while for this case of uncountable infinity it's every number between 0 and 1? But doesn't that not matter when you can map every decimal 1 to 1 with the countable numbers?
You haven't yet. What you're listing isn't "all decimal numbers", but "all decimal numbers that eventually terminate". Numbers like $1/3 = 0.333\ldots$ or $e = 2.182818\ldots$ won't show up in your list.
What your proof shows is that the set of terminating decimals is countably infinite, which is correct!
Conversely, the set of "integers, but you can go infinitely off to the left" is uncountably infinite, as you might now be able to prove.