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

[QuantumQuest]

The problem with the stance described above is in the word "the" in the first sentence. [highlighting mine].

The word "the" carries with it an assertion of existence and uniqueness. Before you are allowed to use it in mathematical discourse, you need to be prepared to justify both existence and uniqueness. I see no possibility for any such justifications.

There is no guest "at infinity" and no need for one.

Well, my main argument is in the first paragraph of post #27.

Now, in the second paragraph, I try to convey some ideas in an essentially informal manner. You are right that the word "the" you've highlighted carries an assertion of existence and uniqueness which, obviously enough, cannot be justified / proved. But, I have put the phrase "at infinity" in separate quotes because there is, exactly, no such mathematical thing as a specific person or a room at infinity. So, "at infinity" is used as a vehicle to serve the idea of each time occupying the next-to-last room and the first room be left unoccupied and nothing more. I have also put the rest of arguments about "infinity" in this same regard, in this paragraph. So, while I know of course the correctness of your argument, what I write is an attempt to convey some ideas in a manner of speaking. I'm really sorry if it was not written in a strict mathematical manner but I think that, reading the whole paragraph, my goal to convey the ideas I want becomes obvious.

 

[Mark44]

But if you turned the tables and ask for the proof of B, it seems like there is no good logical or mathematical proof for B ether, to be shown as true (at least not for me). It feels like the mathematical or logical concepts do not fit for Hilberts Hotel anymore.

You have been repeated shown a logical method for freeing up a single room, one that relies only on the mathematical concept that, for the integers, each integer has a successor -- an integer that is larger by one. Why do you believe that this does not constitute a proof? I think you might have a very narrow idea of what a mathematical proof must be.

So, "at infinity" is used as a vehicle to serve the idea of each time occupying the next-to-last room and the first room be left unoccupied and nothing more.

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...

 

[member 587159]

What is the successor for infinity?

Assuming you mean the ordinal $\omega$ as infinity,the successor is $\omega+1$.

 

[PeroK]

The point is that both A and B are correct and that there is no paradox.

There are infinitely many rooms. All of them are occupied. That is proposition A. Proposition A is correct^*
Room can be made for one more guest without adding any rooms. That is proposition B. Proposition B is also correct^*

One definition for "infinite set" is based on this behavior.

(*) Provided that one is prepared to accept the Peano axioms or many of the various other foundational axiom suites which contemplate infinite sets.

Or, even more succinctly. The function $f: \mathbb N \rightarrow \mathbb N$ such that $f(n) = n+1$ is well defined.

 

[pbuk]

I'm going to try a more direct approach to answering the OP - here is a series of statements which provide an informal proof for the 'always room for 1 more' solution.

1. Room $1$ exists (axiom)
2. If room $n$ exists then so does room $n + 1$ (axiom)
3. At the beginning of a transition all rooms are considered available (axiom)
4. At the end of a transition all rooms may be empty or be occupied by exactly 1 person i.e. full (axiom)
5. Immediately before transition $t_i$ all rooms are full (statement)
6. During transition $t_i$ each person $p_n$ in room $n$ moves to room $n+1$ (proof follows)
   1. Room $n+1$ exists (by 2)
   2. Room $n+1$ is available (by 3)
   3. No other person $p_m, m \ne n$ is moving to room $n+1$ so $p_n$ can move to room $n+1$ (by 4)
7. During transition $t_i$ the person in room 1 moves to room 2 (by 6)
8. After transition $t_i$ room 1 is empty (by 7) QED

 

[QuantumQuest]

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

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.

 

[jbriggs444]

very close to infinity

This is yet another undefined notion.

 

[Mark44]

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.

 

[QuantumQuest]

This is yet another undefined notion.

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

 

[PeroK]

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

How about this:

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?

 

[gmax137]

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.

 

[dakiprae]

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

 

[jbriggs444]

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.

 

[Mark44]

Still struggling, when its ends up in infinity.

It doesn't end up. The number of rooms in this hypothetical hotel is unbounded.

 

[bahamagreen]

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?

 

[jbriggs444]

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]

 

[bahamagreen]

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.

 

[jbriggs444]

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.

 

[Mark44]

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.

It is not polite to pretend to have something relevant to contribute when one does not.

Amen...

 

[bahamagreen]

I'm thinking about it...

 

[gmax137]

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."

 

[dakiprae]

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)

 

[Klystron]

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.

 

[jbriggs444]

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.

 

[dakiprae]

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

 

[Mark44]

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.

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 ...

 

[fresh_42]

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 ...

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]

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

 

[fresh_42]

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.)

 

[256bits]

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.