Infinite hotel +1 guest

Question

Can anyone explain how is it possible to fit another guest in an infinite hotel that is already full?

Here's how the explanation (supposedly) goes:

A guest arrives at an infinite hotel, where every one of the infinite number of rooms is occupied by a guest. The manager tells the guest to go to the first room and ask the person there to move to the next room, and to ask the occupant in the next room to do the same. Thus, they say, the new guest will get accommodation, and since the hotel is infinite, the chain will go on forever.

However, let's look at what is happening step by step:

the hotel is full, there's one potential guest (a) that is not in any room guest knocks on a door, asks the tenant (b) to go to the next room, now there are two guests (a and b) not in any room, and one room is empty guest a goes into the empty room 4 = 1) the hotel is full, there is one potential guest (b) not in any room. Thus there will always be at least one guest not in a room, so we cannot fit another guest into an infinite hotel that is already full, can we?

My question is where is my logic flawed, since it seems that it is commonly understood that you can?

Answer 1

The moving of the guests is not sequential. Or, to keep the with the hotel story, the manager does not go to tenant b to move to the next room. Instead, the hotel manager uses the hotel's computer systems to call all phones in all the rooms at the same time and instructs the guests to all exit their room at the same time, then enter the next room, again all at the same time.

Once all the guests (the infinitely many of them) have done what they were instructed, the manager takes the new guest to room #1 which is now empty.

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Now, of course, you could say "well but the electrical signal for the instructions goes first to room 1, then room 2, and cannot reach all rooms"... but that would be missing a point. Of course it is not physically possible to make the move because it is physically impossible for the hotel itself to exist. That's not the point of the analogy, the point is to show that infinities are strange and sometimes counterintuitive.

The real mathematical message behind the story is that it is relatively easy to translate the story into a rigorous proof that there exists a bijection between a countable set and a countable set to which we add one more element. For example, one can quickly see that there exists a bijection $f$ between $\{1,2,3,\dots\}$ and $\{-1\}\cup\mathbb N$ by defining

$$\begin{align} f(n) = \begin{cases}-1&n=1\\ n+1& \text{otherwise}\end{cases} \end{align}$$

This is counterintuitive, because from finite sets, we are used to the facts that

• if you add one element to the set, then the result is a bigger set
• If you take a proper subset of the set, then that subset is smaller than the set containing it

but this is simply not the case with infinite sets, where two sets that may seem of different sizes are in fact the same size.