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

[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.

 

[pbuk]

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.

 

[srfriggen]

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.

 

[Vanadium 50]

Really, the guests are not numbers

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

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

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

 

[fresh_42]

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.

 

[Buzz Bloom]

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?

 

[phinds]

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.

 

[gmax137]

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.

 

[bob012345]

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.

 

[phinds]

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.

 

[fresh_42]

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.