Magic gains or losses? In Hilbert’s hotel you can mess around with infinity and you end up with magical gains or losses. How can that be?

In the Hotel Hilbert you have an infinite number of occupied rooms. The Hotel is full booked.

In each room is exactly only 1 guest.

A new guest arrives and is asking for a room. The smart receptionist has an idea…

He goes to the first room and tells the guest from room #1 to move on to room #2 and the guest from room #2 also should move on to the next room…..and so on…

By having that said room #1 gets free and he can give it to the new guest.

As you can see we created a new free room in a fully occupied Hotel. Did we?

No. The idea is simple but you do not actual create a new free room or extra space out of infinity…

In fact you borrow it from a extra space not mentioned or you break with the rule that in every room only 1 guest is allowed.

What is happening? 

If you think it over and take a closer look you recognize that once you started this guest movement from now on to the end of time at least one guest is always on the hallway or if you do not have a hallway and the rooms have a direct connection to each other there is always at least 1 room occupied now with 2 guests.

Once this shift is started it runs forever. You can’t finish it. You will never ever have a stable Hotel anymore with a fixed assignment between the guests and rooms. There is always a movement included which never ends.

So it is impossible to get the Hotel from a fixed and stable assignment between the rooms and guests from situation 1 before the new guest arrived into a situation 2 where the new guest got this new room but all other guest have a fixed assignment again. Somebody will always be on the hallway or in another guest’s room.

Taking this into account you see that there is nothing magical going on in the Hotel Hilbert.

Shifting assignments between infinite sets cannot be finished and will always have a disturbance within which cannot be removed. Whatever gain or loss you get by these kinds of shifts you just borrow it from an extra space which you added as silent assumption by starting the shift. 

By not recognizing this silent assumption you end up with unexplained gains or losses. But if you recognize it you see that you did not gain or lose anything. You just borrowed it from an extra space added (Hallway) or by breaking the assignment rule ( 2 guests in one room).

I tried once to make this point clear with a reader comment or letter on an Article I read about the Hotel Hilbert.

I got this analogous answer: But if you could somehow inform all guest simultaneously and move all guests somehow from one room to another then it would work….

So in other words…if I just could make out of a 1 a 2 somehow. Why would I need Hotel Hilbert? If somehow is allowed everything will work.

If I do not need to know anymore how I can do it, every assumption is valid…even 1 = 2. Somehow it could work.

Willi Penker