# Hilbert's Hotel: new Guest arrives (Infinite number of Guests)

• B
• dakiprae
In summary, Hilbert's Hotel is a thought experiment that explores the concept of infinity. It describes a hotel with an infinite number of rooms, each occupied by exactly one guest. When a new guest arrives, the current guests can be shifted to the next room and make room for the new guest. This process can be repeated to accommodate any finite number of new guests. However, there is no proof that a single new guest can be moved into room 1, as it is an axiom that every natural number has a successor. While this concept may seem impossible in reality, it is mathematically possible.
Mark44 said:
There is no next-to-last room, any more than there is a last room. "At infinity" doesn't apply here. In a similar vein, there is also no last '9' digit in the number 0.999...

Fair enough. Although I just try to give an abstract idea - I have already mentioned in post #27

QuantumQuest said:
As has already been pointed out, there is no next to infinity as there is also no previous to it; in other words you can't add or subtract or do any other math operations at infinity.

I think that I did it in a rather violating way regarding math lingo (referring also to post #28 by @jbriggs444). So, I apologize for it. What I essentially mean is that our counting goes till very close to infinity (not "at infinity" as I said).Now, let me give it in a more formal manner as a continuation of the first paragraph of post #27.

We use the bijection ##f(n) = n + 1## in order to relocate all guests. This holds for the simple variation of one guest arrives each time. For the other variants of the problem we can also create an appropriate bijection. The whole idea is that we can put a set having infinitely many elements into one-to-one correspondence with (any) one of its proper subsets.

QuantumQuest said:
very close to infinity
This is yet another undefined notion.

QuantumQuest said:
We use the bijection ##f(n) = n + 1## in order to relocate all guests. This holds for the simple variation of one guest arrives each time. For the other variants of the problem we can also create an appropriate bijection. The whole idea is that we can put a set having infinitely many elements into one-to-one correspondence with (any) one of its proper subsets.
Yes. If we need to find space for 2 arriving guests, the bijection ##f(n) = n + 2## will do, and similar for any finite number N of new arrivals.
If we need to find space for a (countably) infinite number of new guests, here's a bijection that will work: ##f(n) = 2n##. After the existing guests move, they will have moved to rooms with even numbers, freeing up all of the odd-numbered rooms.

Klystron and QuantumQuest
jbriggs444 said:
This is yet another undefined notion.

What I mean is: up to the point where we have no means whatsoever to count further.

weirdoguy
QuantumQuest said:
Fair enough. Although I just try to give an abstract idea - I have already mentioned in post #27

You are working reception at the Hilbert Hotel, which is full. A new guest arrives and you issue the order for every guest to move to the next room; leaving room 1 vacant for the new guest.

The phone rings because two people are now having to share. From which room is the phone call coming?

I thought the point of the Hilbert Hotel was more about understanding the meaning of "infinity" than it is about hotels or rooms or proofs.

Say you see a billboard advertisement for a hotel, "We have an infinite number of rooms!" So you call and ask, "Is there always room for another guest?" If the clerk says "no, once every room is occupied we can take no more" then you know the billboard is just advertising hype.

I think there are good reasons out there for the n + 1 peano axioms. Still struggling, when its ends up in infinity. I am going to leave this topic for now. Thank you all for your input

QuantumQuest said:
What I mean is: up to the point where we have no means whatsoever to count further.
That is yet another undefined notion. There is no such point in the natural numbers. By axiom, each one has a successor.

weirdoguy
dakiprae said:
Still struggling, when its ends up in infinity.
It doesn't end up. The number of rooms in this hypothetical hotel is unbounded.

Let the room numbers and occupants be unique and paired natural numbers such that:
1 is in room 1, 2 is in room 2, 3 is in room 3...
Is there any question that the hotel now contains all of the natural numbers?
If posed a unique natural number not yet a guest shows up, what is this number?

bahamagreen said:
Let the room numbers and occupants be unique and paired natural numbers such that:
1 is in room 1, 2 is in room 2, 3 is in room 3...
Is there any question that the hotel now contains all of the natural numbers?
If posed a unique natural number not yet a guest shows up, what is this number?
Let the new guest be assigned a name instead: "new guest" such that "new guest" is not an element of ##\mathbb{N}##

Is there a possible bijection between ##\mathbb{N} \cup \{\text{new guest\}}## and ##\mathbb{N}##?

[I find it distasteful to dodge the problem by saying that no new guest shows up]

Considering a kind of object, looks to me like "all" might be finite or infinite, and "infinite" might be some or all.

With natural numbers 1, 2, 3..., it looks to me like "all" does mean infinite and infinite does mean all. So no new guests in the form of a natural number.

I'm not seeing a bijection; {new guest} is not paired with a natural number. But I'm not seeing the motivation to propose a different kind of object as the new guest.

weirdoguy
bahamagreen said:
Considering a kind of object, looks to me like "all" might be finite or infinite, and "infinite" might be some or all.
"All" is a quantifier. Not an object.
With natural numbers 1, 2, 3..., it looks to me like "all" does mean infinite and infinite does mean all.
That is not correct. There are many infinite subsets of the natural numbers. Uncountably many. Only one of those subsets consists of all of the natural numbers.
I'm not seeing a bijection; {new guest} is not paired with a natural number.
Yes. It is. The bijection that is proposed maps "new guest" to 1, 1 to 2, 2 to 3 and so on.
But I'm not seeing the motivation to propose a different kind of object as the new guest.
If you have no new guests to come to the hotel, the question of where to put a new guest does not arise. Insisting the new guest be a natural number would be dodging the scenario. It is not polite to pretend to have something relevant to contribute when one does not.

phinds, Klystron and Mark44
bahamagreen said:
If posed a unique natural number not yet a guest shows up, what is this number?
What does this even mean? It looks like something that Yoda might say, except that it's completely unintelligible.

jbriggs444 said:
It is not polite to pretend to have something relevant to contribute when one does not.
Amen...

Really, the guests are not numbers; we just use the numbers as their names, or a code for their names.

New guest to clerk: "Hello, I'm Mister Zero, do you have a room available tonite?"
Clerk: "Why, I'm sure we can find you a room."

New guest to clerk: "Hello, I'm Mister Onepointfive, do you have a room available tonite?"
Clerk: "Why, I'm sure we can find you a room."

New guest to clerk: "Hello, I'm Mister Mxyzptlk, do you have a room available tonite?"
Clerk: "Why, I'm sure we can find you a room."

gmax137 said:
Really, the guests are not numbers; we just use the numbers as their names, or a code for their names.

The thing is, if there could exist infinite rooms (represented by number 1,2,3...), you have to belief the rooms (numbers) just exist. If you can multiply all numbers by *2, then obviously not all numbers existed before. But if that multiplication works, you still have infinite free rooms and infinite occupied rooms left. I had no problem with that math or logic, that every number goes n+1 or n*2 or moving to prime numbers. The whole example just going over my imagination. That is why I say, I have no idea what would going on with this hotel (if it really could exist)

weirdoguy
dakiprae said:
The whole example just going over my imagination. That is why I say, I have no idea what would going on with this hotel (if it really could exist)
As @fresh_42 and others have explained, the "hotel with guests" metaphor represents sets. If you understand the set of natural numbers, then the "hotel" has accomplished its purpose as an aid to understanding.

fresh_42
dakiprae said:
If you can multiply all numbers by *2, then obviously not all numbers existed before
If you want to find the error in a mathematical argument, look for the words "clearly" or "obviously".

The set of all natural numbers contains all of the natural numbers, including the ones you get when you multiply them all by two.

phinds, etotheipi, PeroK and 2 others
jbriggs444 said:
If you want to find the error in a mathematical argument, look for the words "clearly" or "obviously".

The set of all natural numbers contains all of the natural numbers, including the ones you get when you multiply them all by two.

Yes you right, that was a bad example

dakiprae said:
That is why I say, I have no idea what would going on with this hotel (if it really could exist)
Obviously, a hotel with an infinite number rooms can't exist. The Hilbert Hotel is purely a thought experiment.
jbriggs444 said:
If you want to find the error in a mathematical argument, look for the words "clearly" or "obviously".
This is good advice most of the time ...

Klystron and jbriggs444
jbriggs444 said:
If you want to find the error in a mathematical argument, look for the words "clearly" or "obviously".
Mark44 said:
This is good advice most of the time ...
Indeed.

I like the version: As can easily be seen ... If it is so easy, obvious or clear, why the heck don't you write it down. This is especially true on the internet, outside of textbooks. Weaknesses are frequently hidden behind such phrases.

etotheipi and Klystron
fresh_42 said:
I like the version: As can easily be seen ... If it is so easy, obvious or clear, why the heck don't you write it down. This is especially true on the internet, outside of textbooks. Weaknesses are frequently hidden behind such phrases.

Reminds me of this:
One day Shizuo Kakutani was teaching a class at Yale. He wrote down a lemma on the blackboard and announced that the proof was obvious. One student timidly raised his hand and said that it wasn’t obvious to him. Could Kakutani explain? After several moments’ thought, Kakutani realized that he could not himself prove the lemma. He apologized, and said that he would report back at their next class meeting.

After class, Kakutani, went straight to his office. He labored for quite a time and found that he could not prove the pesky lemma. He skipped lunch and went to the library to track down the lemma. After much work, he finally found the original paper. The lemma was stated clearly and succinctly. For the proof, the author had written, “Exercise for the reader.” The author of this 1941 paper was Kakutani.

Steven G. Krantz, Mathematical Apocrypha: Stories and Anecdotes of Mathematicians and the Mathematical

phinds, DrClaude, jim mcnamara and 3 others
etotheipi said:
Reminds me of this:
The honest version which usually doesn't hide something is: "straight forward calculation". This is normally true and can easily be done, e.g. an induction. At least it says, that one doesn't have to look for complicated tricks.

I once had an "obvious" on a transformation of complex numbers which took me three days and several substitutions (change of variables) to figure out. And some errors are hidden in plain sight: ##a<b \Longrightarrow ac<bc##. If ##a,b,c## are complicated expressions, who cares that ##c<0## couldn't be ruled out? (Seen in a PhD thesis.)

etotheipi
Mark44 said:
What does this even mean? It looks like something that Yoda might say, except that it's completely unintelligible.
Or Buzz Lightyear "To infinity and beyond!"

Anyways,
How did the hotel fill up in the first place..
There is a convention in town and new The Infinite Hotel is open for business.
Guests arrive and keep on arriving. In fact there is an infinite number of them.
As they arrive in the lobby they check in and are assigned guest / room 1,1 . 2,2 , .. m, n, ... , ...
The hotel management always sees an infinite number of guests in line, and they always have a room available.

256bits said:
Or Buzz Lightyear "To infinity and beyond!"

Anyways,
How did the hotel fill up in the first place..
There is a convention in town and new The Infinite Hotel is open for business.
Guests arrive and keep on arriving. In fact there is an infinite number of them.
As they arrive in the lobby they check in and are assigned guest / room 1,1 . 2,2 , .. m, n, ... , ...
The hotel management always sees an infinite number of guests in line, and they always have a room available.
Well this is a different question - in the OP you start with the hotel full and have to squeeze in one more guest.

But it's not worth starting a new thread to answer your side question - the hotel can easily fill up if the first guest takes 1 minute to check in, the second guest 30 seconds, then 15, 7.5 etc. After 2 minutes there are no more guests in the queue and every room is occupied.

256bits
fresh_42 said:
Interesting physics question here: The communication of the order to move is of finite speed. Whereas this doesn't seem to be a problem for the first billion rooms, will it work out at infinity?
That's a long game of "telephone." The billionth guest would hear, "Purple baby monkey uncle" and have no idea what to do.

PeroK
gmax137 said:
Really, the guests are not numbers

I am not a number! I am a free man!

fresh_42 said:
I like the version: As can easily be seen ...

I like "One can show". One did. His name was probably Gauss.

I like "One can show". One did. His name was probably Gauss.
Or Euler. I wonder whether it was Fermat who started to write like this.

I think the OP has a point related to the distinction between a math problem and a real world problem. There is no constraint that axioms related to a math problem must reflect reality. If we make an assumption that there are an infinite number of rooms, say all in a line with a shared corridor, how long does it take for the message to be passed to all of the rooms that each occupant has to move to the room next door? How can it be done in less than an infinite amount of time?

Buzz Bloom said:
I think the OP has a point related to the distinction between a math problem and a real world problem.
I disagree. It seems to me that the OP is trying to force a math problem to look like a real world problem and using that false comparison to confuse himself about what really is just a math problem. His mistake is understandable since it is, unfortunately, POSED as a real-world problem but only a beginner would try to interpret it as actually BEING one.

fresh_42, etotheipi and Mark44
I think 'the Hilbert Hotel' is a teaching tool to help you understand what "infinity" is about; what it means to have a set with "an infinite number" of members.

Is it weird? yes, but that's the point.

DrClaude
fresh_42 said:
Or Euler. I wonder whether it was Fermat who started to write like this.
All I can say is that it's uncertain if Heisenberg ever checked into the Hilbert Hotel. But Schrödinger was probably spread over several rooms. But all this is getting rather Bohring.

bob012345 said:
All I can say is that it's uncertain if Heisenberg ever checked into the Hilbert Hotel. But Schrödinger was probably spread over several rooms. But all this is getting rather Bohring.
Yes, even aside from the pun, it is, and based on the fact that the whole conversation started based on a misunderstanding by the OP, it seems to me it should be closed, as is normally done here on PF with such threads.

phinds said:
Yes, even aside from the pun, it is, and based on the fact that the whole conversation started based on a misunderstanding by the OP, it seems to me it should be closed, as is normally done here on PF with such threads.
Guess this is a good idea.

Hilbert's hotel is a heuristic, no mathematical construction.

• General Math
Replies
8
Views
2K
• General Math
Replies
73
Views
4K
• Set Theory, Logic, Probability, Statistics
Replies
7
Views
2K
• General Math
Replies
27
Views
3K
• General Math
Replies
1
Views
1K
• General Math
Replies
31
Views
5K
• Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
• General Math
Replies
18
Views
4K
• Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
9K