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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?

Asaf Karagila
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DePianoman
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    You're only including the terminating decimal numbers. What about $1/3$, $\sqrt2$, $\pi$, etc.? – Hans Lundmark Apr 27 '22 at 05:29
  • What if I were to argue there are non-terminating natural numbers just by removing the decimal point? That number still exists, no? When counting upwards would you not reach those numbers? Or is it just a number that does not exist, which I don't think is the case? – DePianoman Apr 27 '22 at 05:33
  • Again, not trying to play Devil's Advocate, just trying to understand – DePianoman Apr 27 '22 at 05:34
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    Sorry, I don't understand. There's no such thing as a “non-terminating natural number”. Any natural number has a finite number of digits. – Hans Lundmark Apr 27 '22 at 05:36
  • I guess the big disconnect is I don't see how non-terminating decimals are any different from infinitely long natural numbers, and that's just a part of this section of math that I haven't seen yet. Again, just getting into this field and this was something I noticed and immediately saw as something I have a misunderstanding about. I'll take another read through the article listed in Henry's answer and see if I can grasp the concept better. – DePianoman Apr 27 '22 at 05:39
  • The key is precisely that the jump from "only finitely many nonzero digits" to "possibly infinitely many nonzero digits" is where we jump from "countable" to "uncountable". – Arturo Magidin Apr 27 '22 at 05:46
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    The phrase “infinitely long natural numbers” is self-contradictory. You've padded your natural numbers with infinitely many zeros on the left, but that doesn't change the fact that their digits actually “stop” at some point. If you pad on the left with a string of infinitely many digits that are not all zero, what you get is no longer a natural number. – Hans Lundmark Apr 27 '22 at 05:47
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    Ah, that makes sense then yeah. I guess the big disconnect was: "So if the series is infinite, why doesn't in include numbers that are infinitely long?" Thinking about it it does make sense that that wouldn't be included as when counting it really wouldn't make sense that you would reach infinitely many numbers. Again, new to the topic and wrapping my head around infinities is definitely a daunting and confusing idea at the moment. Thanks for clarifying though! – DePianoman Apr 27 '22 at 05:49
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    By the way, lots of similar questions have been asked on this site before: https://www.google.com/search?q=site:math.stackexchange.com+natural+number+infinitely+many+digits – Hans Lundmark Apr 27 '22 at 05:52

1 Answers1

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Why doesn't this system work, and at what point did I change it from countable to uncountable?

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.

Henry Swanson
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