I'm not a mathematician at all. From this I read a sentence like this :
A. Imagine a Grand Hotel with a (countably) infinite number of floors and
rooms. On this particular night, the hotel is completely full.
The thing that I don't understand is :
Why the illustration say "the hotel is completely full" ?
When we hear "the hotel is completely full" doesn't that mean this hotel has a finite number of rooms ?
The article continue :
B. Late in the evening, you arrive at the hotel and inquire about a room.
Although there is no vacancy the hotel manager tells you that since
this is an infinite hotel she can easily make room for you!
To me, it seems the illustration want to show that the hotel will make one room each time there is a guest to check in. So, if at 3 pm there are 10 guests want to check in, then the hotel make 10 rooms. By 4 pm, there are another guest want to check in, then the hotel make one more room. Please correct me if I'm wrong.
BUT, the article continues :
C. You’re delighted, but confused. If the infinite hotel is completely
filled with an infinite number of guests, how does the manager go
about securing a room for you?
I don't understand, why the illustration say that I'm confused ?
If I know that the hotel can always receive a guest just before I check in then why do I confuse ?
The thing that I confuse is if the manager tell me "our hotel is completely full" at the time I want to check in. So, it's not the "how does the manager go
about securing a room for me" which made me confuse, but why the hotel's manager told me "our hotel is completely full" which made me confuse.
The article continue :
D. Now this is perplexing. At first you may think, “Well since it’s
infinite can’t we just put the new guest in the last room?”
To me, the thing that "at first I may think" is :
How come it is said "the hotel is completely full" ?
And of course, since it's infinite – there will never be "a last room".
To me, the hotel will never say "we are completely full" since it has an infinite number of rooms.
Except, "the hotel is completely full" means at the time I want to check in. But then there is no problem since the hotel can always make another new room for each of their future guests as seen in point B. So, "the last room" means the newly build one – not the existing one. The hotel's manager doesn't have to move each existed guest to the next room in order to put me in room #1, but just put me to the newly build one, "the last room".
What did I do wrong in understanding this Hilbert's hotel illustration ?
I'm sorry as my question above is not fit to be asked. Now I understand that it seems the answer to my last question : "What did I do wrong in understanding this Hilbert's hotel illustration ?" is simple —> "I put the illustration as if in a real event/condition" which actually I should try to put it in a mathematical condition.
I'm sorry once again for my mistake :).
Thank you for all the answers and the comments.
I give up.
Best Answer
A. We label the hotel rooms in $\mathbb{N}$, that is in terms of $1, 2, 3, \ldots$ and the hotel is full in the sense that whenever you give me a positive room number, I can go check that room and there will be a customer there. Hence, there is no empty room.
B. Now, you just arrive at the hotel and you want to check in. hmmm... each room is occupied, but you will be accomodated by making the customer in room $i$ to room $i+1$ and the first room is assigned to you. Hence a room is given to you (i.e. a room is made in this sense). It doesn't mean we build a new room but we just shift the customers.
C. The description that you are confused perhaps treat the new customer as someone who is new to the idea of countably infinite.
D. I believe part $A$ has answered your question that the hotel can be full. And yes, there is no such thing as the last room in this context just like there is no such thing as the biggest positive integer. Also, the writing treat it as someone is not familiar with countably infinite.