Can I have visited all rooms of Hilbert's infinite hotel?

Question

Thanks in advance for any helpful comments! As this is all theory and abstract, please don't leave comments about the practicality of this (unless you feel that it is really relevant). Comments like "infinity is an impossibility anyway" don't really help here. Some people accept and some reject infinity. So for the sake of this post please let's all accept it for a moment.

Here I'll use Hilbert's Grand Hotel paradox to try to illustrate my question: Lets say there was a Hotel with infinitely many rooms. And let's say that every day, I decide to change to the room to the left of me (and there are infinitely many rooms to my left). Every day I get into a new room. Now, let's suppose I have been doing this for an INFINITE amount of time. For infinitely many days have I been changing the room into a new room of the hotel with infinitely many rooms. Does this mean I have been in ALL rooms? If I have been doing this for infinitely many days, would I still have infinitely many rooms to visit that I didn't visit yet? Because I'm thinking, sure, you can never reach the end of infinity. BUT, the infinity of how many rooms there are is the same size of infinity as how many days I have been staying at the hotel, right? And to my logic, in order for me to NOT have visited all rooms, I would have to not have stayed at the hotel for an infinite amount of days.

If I have visited the same amount of rooms as there are rooms, so I have visited infinitely many rooms in an hotel with infinitely many rooms, I would have to have visited all rooms, right? And second question: Is there any issue with "sizes of infinity" in this question? Like, am I maybe missing something that indicates that the sizes of the two infinities (infinitely many rooms and infinitely many days) do actually differ? Because I'm thinking, hotel Rooms are counted like 1, 2, 3, 4... and so are days.

Please note that wrote that I already HAVE BEEN staying at the hotel for an infinite amount of days. Not that I am staying at the hotel for an infinite amount of days. Not sure if it makes a difference, but I still wanna point it out.

Again, thanks in advance for any useful answers!

Answer 1

There are a number of frustrating things with the question of what happens after having done something for an infinite amount of time.

Consider the hypothetical scenario that the hotel is continuing to be built at a rate of one room per day and you are changing rooms once per day and you stay in the first room the day it is built. The number of total rooms minus the number of used rooms will give the number of unused rooms. In this scenario we have $0$ unused rooms every day. As the number of days approaches infinity, the limit of the number of unused rooms will remain zero. That is to say

$$\lim\limits_{n\to \infty} (n-n)=0$$

Consider a different scenario where each day two rooms are built and you stay in this first room on the first day. Each day, the number of unused rooms increases and on the $n$'th day there will be $n$ unused rooms. The limit as the number of days increases of the number of unused rooms here is infinity. That is to say

$$\lim\limits_{n\to\infty}(2n-n)=\infty$$

Consider a third scenario where each day one room is built but you only begin staying in the hotel on the day the third room is built, again staying in only one room a day. Here, after $n$ days of you staying in the hotel there will continually be $2$ rooms you haven't visited, so the limit of the number of unused rooms as $n$ approaches infinity will be $2$. That is to say

$$\lim\limits_{n\to\infty}((n+2)-n)=2$$

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The question you seem to be asking is in effect, what is $\infty-\infty$? Under normal circumstances, we would hope that $\lim(a_n-b_n)=\lim(a_n)-\lim(b_n)$ but in each of the above examples, if we were to have split these apart we would arrive at one of these $\infty-\infty$ situations.

$0=\lim\limits_{n\to\infty}(n-n)=^*\lim\limits_{n\to\infty}n - \lim\limits_{n\to\infty} n=\infty-\infty$

$\infty = \lim\limits_{n\to\infty}(2n-n)=^*\lim\limits_{n\to\infty}2n - \lim\limits_{n\to\infty} n = \infty-\infty$

$2=\lim\limits_{n\to\infty}((n+2)-n)=^*\lim\limits_{n\to\infty}(n+2)-\lim\limits_{n\to\infty}n = \infty-\infty$

When you say the hotel has infinitely many rooms and you've been staying for an infinite amount of time, as far as well can tell you are asking for a value of $\lim\limits_{x\to\infty}x - \lim\limits_{y\to\infty}y=\infty-\infty$ which does not have a unique answer.

Further, it isn't clear how we should treat these infinities, whether they should be defined using limits in the first place or whether they should be treated as numbers themselves. If the infinity of the number of rooms isn't defined using limits but the infinity of the number of days you have stayed at the hotel is then that doesn't seem fair.

What we can say however, is that assuming you do not skip over any rooms, there will have been a day where you have visited room number $n$ for any finite value of $n$, so you are incapable of pointing to a specific room which you will have not yet visited eventually. In that sense, if I had to argue towards any specific answer, it would be that you will have visited every room $(*)$ but I still say the question itself is flawed to begin with.

Answer 2

One definition of infinite is: A set is infinite if it can be put in one-to-correspondence with a proper subset of itself.

No finite set has this property. If you try to match all the natural numbers up to $n$ with themselves in a 1-1 manner, no matter how you do it, there will never be any left over. This is something we observe constantly. All of our experience with matching things supports this, and our intuition naturally assumes it. But when dealing with the infinite, this breaks down. Our intuition misleads us. We have to painstaking develop a new intuition for the infinite. This is something every student of higher mathematics has to go through anymore. Hilbert came up with his "heavenly hotel" exactly for the purpose of guiding others (and perhaps himself) in developing that new intuition.

Where your intuition is misleading you is here:

Because I'm thinking, sure, you can never reach the end of infinity. BUT, the infinity of how many rooms there are is the same size of infinity as how many days I have been staying at the hotel, right? And to my logic, in order for me to NOT have visited all rooms, I would have to not have stayed at the hotel for an infinite amount of days.

This idea only makes sense for finite sets. If you match them up 1-1 and have some left over on one side, then that side must be larger than the other, right? When dealing with the infinite, that is wrong.

In your example, you say you always moved left. But what about all the rooms to the right of the one you started in? You will never visit them. But hey, if you start in room 0, there are no rooms to the right, so you visited them all!. But wait, you got room 0, and your buddy was in room 1, and he did the exact same thing you did. But he never got to visit room 0. And Auntie Freda was in room 23 and did it too. She did the exact same thing you did, but missed 23 rooms.

And then you go on the next hotel and do the same thing there. But this time, they have odd numbered rooms on one side of the hall, and even numbered on the other. By moving left each time, you stayed on the even side of the hall and never visited any of the odd rooms. So this time - though you did the exact same thing, you missed an infinite number of rooms.

At the third hotel, things are even worse. There are no room numbers! And the halls are all confusing so there is no way to just move left. So you pick a room at random, but leave a mark in it, so that you know to never visit it again. After your infinite number of days, have you visited them all? No one knows. It is impossible to say.

With the infinite, you can match your infinite number of rooms with your infinite number of days and have any number from 0 to infinite left over. This is something you just have to get used to.

Answer 3

If I have visited the same amount of rooms as there are rooms, so I have visited infinitely many rooms in an hotel with infinitely many rooms, I would have to have visited all rooms, right?

No. For example, maybe there are also infinitely many rooms to your right, none of which you ever visit.

The short answer to your larger question is that it depends on 1) how the rooms are laid out in the hotel and 2) given how the rooms are laid out, the exact sequence in which you visit them. There are many strange things that can happen. For example, let's think about placing the rooms at different points on the real number line. Suppose there are rooms placed at the points

$$0, \frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \dots 1 - \frac{1}{2^n}$$

for all positive integers $n$. Then it is possible to visit all of these rooms by starting at $0$ and moving to the room with the next largest coordinate each time. But now suppose there are also rooms at the points

$$1, 1 + \frac{1}{2}, 1 + \frac{3}{4}, \dots 2 - \frac{1}{2^n}.$$

Then after infinitely many days, my original strategy still misses infinitely many rooms! In other words, there can still be infinitely many rooms even further in the positive direction, even though I've moved in the positive direction infinitely many times.

On the other hand, there is a different strategy which will result in me visiting every room: namely, I alternate between the two sets of rooms, first visiting $0$, then $1$, then $\frac{1}{2}$, then $1 + \frac{1}{2}$, and so forth.

Answer 4

You say you keep moving to the room to the left. So I assume there are infinitely many rooms to the left of your starting point. Will visiting all those rooms mean you will visit all rooms? No, because there can also be infinitely many rooms to the right (or going north, south, up a level, down a level, etc etc.).

In short: just because you visit infinitely many rooms does not mean you visit all rooms, even if there are 'just' countably many rooms. For another example of this: if the rooms are numbered 1,2,3,..., and you visit rooms 2,4,6,..., then you visit infinitely many rooms, but obviously not all rooms.

Put differently yet: you can visit 'the same amount' of rooms as there are rooms, and still not visit all rooms ... where 'the same amount' is understood as equal cardinality of course.