Hilbert's paradox of the Grand Hotel - An easier solution?

[Lars Krogh-Stea]

TL;DR Summary

Why the differen schemes of moving people around? When new people arrive, why not make everyone go outside, ande then make them all go in and pick a room. There should be infinite rooms for them to pick from, and everyone gets a room.

I know the paradox is about a hypotethical hotel with hypotethical guests, how else could new people arrive, when everyone (infinite) is already inside;) But maybe the ones outside are aliens, I don't know..
My real point though, is that there's no need for the intricate shuffeling of guests. When new guest arrive, just tell all the guest that are already in the hotel to step outside (or come to the lobby, I guess that's of infinite size too). Now you have an empty hotel with infinite rooms, and infinite people outside. Just tell them all to go inside and choose a room as there are infinite rooms to choose from. Does it need to be more complicated?

 

[PeroK]

Summary:: Why the differen schemes of moving people around? When new people arrive, why not make everyone go outside, ande then make them all go in and pick a room. There should be infinite rooms for them to pick from, and everyone gets a room.

I know the paradox is about a hypotethical hotel with hypotethical guests, how else could new people arrive, when everyone (infinite) is already inside;) But maybe the ones outside are aliens, I don't know..
My real point though, is that there's no need for the intricate shuffeling of guests. When new guest arrive, just tell all the guest that are already in the hotel to step outside (or come to the lobby, I guess that's of infinite size too). Now you have an empty hotel with infinite rooms, and infinite people outside. Just tell them all to go inside and choose a room as there are infinite rooms to choose from. Does it need to be more complicated?

What if it's raining outside?

 

[Mark44]

I know the paradox is about a hypotethical hotel with hypotethical guests, how else could new people arrive, when everyone (infinite) is already inside;) But maybe the ones outside are aliens, I don't know..
My real point though, is that there's no need for the intricate shuffeling of guests. When new guest arrive, just tell all the guest that are already in the hotel to step outside (or come to the lobby, I guess that's of infinite size too). Now you have an empty hotel with infinite rooms, and infinite people outside. Just tell them all to go inside and choose a room as there are infinite rooms to choose from. Does it need to be more complicated?

You're missing the whole point; namely, that even though there are currently an infinite number of people in the hotel's rooms when the new bunch (infinitely many) arrives, the scheme of asking the current guests to move to a room whose number is two times their old room number, clearly makes room for the new guests. An existing guest in room N is asked to go to room 2N. I don't see that as being an intricate shuffling.
A flaw in your method is that it doesn't specify where the existing guests are supposed to go, and where the new arrivals are supposed to go.

The underlying mathematics of Hilbert's Hotel is how one can tell that two infinite sets are the same size.

 

[Lars Krogh-Stea]

You're missing the whole point; namely, that even though there are currently an infinite number of people in the hotel's rooms when the new bunch (infinitely many) arrives, the scheme of asking the current guests to move to a room whose number is two times their old room number, clearly makes room for the new guests. An existing guest in room N is asked to go to room 2N. I don't see that as being an intricate shuffling.
A flaw in your method is that it doesn't specify where the existing guests are supposed to go, and where the new arrivals are supposed to go.

The underlying mathematics of Hilbert's Hotel is how one can tell that two infinite sets are the same size.

I think you're missing my point.. It's hard to see why the groups of infinite people would be anything other than the same size, as any subset of an infinite number (however you shuffle the room, even if you use primes) will be infinite and therefore cardinally the same.
My point is that it's not a paradox. Which is proven by the easy solution i proposed. I mean why wouldn't a hotel with infinite rooms fit infinite people? Or, more precisely, why is it often presented as a riddle, with a complicated answer, when the "practical" (in the hypotethical sense) solution is extremely simple.

It's a bit like the Monty Hall problem, which really isn't a math problem. The math almost couldn't be simpler, it's the psychology of it that makes it hard.

 

[PeroK]

Or, more precisely, why is it often presented as a riddle, with a complicated answer, when the "practical" (in the hypotethical sense) solution is extremely simple.

It's presented as a riddle because, perhaps, most people are not familiar with Cantor's theory of transfinite numbers. Don't forget that Cantor's work was controversial when he published:

https://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory#Reception_of_the_argument

But, yes, the solution is simple to anyone with knowldege of modern set theory.

 

[Office_Shredder]

Given Lars' post, I assume they do not realize if you brought a bus with one person for every real number in it, that those people could not fit in the hotel.

 

[PeroK]

Given Lars' post, I assume they do not realize if you brought a bus with one person for every real number in it, that those people could not fit in the hotel.

If you built a hotel with a room for every real number, what would you put on each door to identify the room?

 

[Lars Krogh-Stea]

Given Lars' post, I assume they do not realize if you brought a bus with one person for every real number in it, that those people could not fit in the hotel.

What entity number would the pepople already in the the hotel have? I'm not sure that bus would be even hypothetically possible ;)

 

[valenumr]

Infinity is not a number. It is a concept. The continuum hypothesis is a good example of how to think about infinite sets.

 

[PeroK]

Infinity is not a number. It is a concept.

A number is a concept, too.

https://en.wikipedia.org/wiki/Aleph_number

 

[Mark44]

My point is that it's not a paradox. Which is proven by the easy solution i proposed. I mean why wouldn't a hotel with infinite rooms fit infinite people? Or, more precisely, why is it often presented as a riddle, with a complicated answer, when the "practical" (in the hypotethical sense) solution is extremely simple.

The paradox is that the hotel is full (each of the infinite number of rooms has someone in it), but we can still fit an infinite number of people into their own rooms. To people unfamiliar with transfinite numbers (which is probably the majority of people on earth), this is a paradox.

The "practical" solution is the usual one; namely where each existing guest moves from room N to room 2N. This scheme exhibits the isomorphism that demonstrates that the two sets--the set of positive integers and the set of even positive integers--have the same cardinality.

In the solution you described, it's impossible to say where a given guest has to move, so this scheme doesn't demonstrate the isomorphism between the two sets.

 

[valenumr]

A number is a concept, too.

https://en.wikipedia.org/wiki/Aleph_number

Fair enough, but at least algebraic numbers are somewhat concrete. The question is posed in terms of natural numbers.

 

[PeroK]

My real point though, is that there's no need for the intricate shuffeling of guests. When new guest arrive, just tell all the guest that are already in the hotel to step outside (or come to the lobby, I guess that's of infinite size too). Now you have an empty hotel with infinite rooms, and infinite people outside. Just tell them all to go inside and choose a room as there are infinite rooms to choose from. Does it need to be more complicated?

That doesn't work, as whether you get everyone back into the hotel depends on how you organise things. For example, if we have an infinite number of women and men and we decide that the ladies should be accommodated first, then none of the men ever gets a room.

 

[Lars Krogh-Stea]

Infinity is not a number. It is a concept. The continuum hypothesis is a good example of how to think about infinite sets.

I think the point here, is to separate countably infinite sets from uncountable sets. With uncountable sets, like the bus with a passenger for every real number, it wouldn't be possible (in my limiteted knowledge) to use a bijective function to map the passengers to the set of hotel rooms.

 

[valenumr]

Fair enough, but at least algebraic numbers are somewhat concrete. The question is posed in terms of natural numbers.

Ugh that was bad phrasing. What maybe a better term would be "worldly". Most people grasp the concept.

Take for example transcendental numbers like pi or e. Does an integer exist with infinite numbers that actually equals one or the other if you move the decimal to the right place?

 

[Lars Krogh-Stea]

That doesn't work, as whether you get everyone back into the hotel depends on how you organise things. For example, if we have an infinite number of women and men and we decide that the ladies should be accommodated first, then none of the men ever gets a room.

I think that's included in the "hypothetical part". Or else the hotel wouldn't be full in the first place. It's like the paradox where you would never reach the door, because first you need to cover half the distance etc.. In the same way you just get up and walk over to the door, you just tell people to pick a room and they do ;)

 

[valenumr]

The paradox is that the hotel is full (each of the infinite number of rooms has someone in it), but we can still fit an infinite number of people into their own rooms. To people unfamiliar with transfinite numbers (which is probably the majority of people on earth), this is a paradox.

The "practical" solution is the usual one; namely where each existing guest moves from room N to room 2N. This scheme exhibits the isomorphism that demonstrates that the two sets--the set of positive integers and the set of even positive integers--have the same cardinality.

In the solution you described, it's impossible to say where a given guest has to move, so this scheme doesn't demonstrate the isomorphism between the two sets.

You could just move tenants up one room and add guests one at a time. If the hotel has infinite rooms, infinity plus one is still infinity.

 

[valenumr]

I think that's included in the "hypothetical part". Or else the hotel wouldn't be full in the first place. It's like the paradox where you would never reach the door, because first you need to cover half the distance etc.. In the same way you just get up and walk over to the door, you just tell people to pick a room and they do ;)

Can something with infinite capacity be full?

 

[Lars Krogh-Stea]

Can something with infinite capacity be full?

Seems like it can't :)

 

[Office_Shredder]

I think that's included in the "hypothetical part". Or else the hotel wouldn't be full in the first place. It's like the paradox where you would never reach the door, because first you need to cover half the distance etc.. In the same way you just get up and walk over to the door, you just tell people to pick a room and they do ;)

It is still the case that if there was one person for every real number, this wouldn't work

The Zeno's paradox reveals a deep truth about the convergence of infinite sums, and the hotel is intended to reveal deep truths about infinity. Dismissing it as 100% trivial is only going to keep you from learning things

 

[valenumr]

Seems like it can't :)

Did you mean Cant(or)

 

[Mark44]

You could just move tenants up one room and add guests one at a time. If the hotel has infinite rooms, infinity plus one is still infinity.

Sure, but this would take an infinite amount of time, assuming some nonzero amount of time for the moves. If all the current guests move at the same time, then all the newcomers can take possession of the newly vacant rooms.

Can something with infinite capacity be full?

Sure, if every room has a (current) guest in it.

Seems like it can't :)

See above.

 

[valenumr]

Id

Sure, but this would take an infinite amount of time, assuming some nonzero amount of time for the moves. If all the current guests move at the same time, then all the newcomers can take possession of the newly vacant rooms.

Sure, if every room has a (current) guest in it.

See above.

I'm not sure how time comes into play with the thought experiment. I hadn't thought about that previously.

 

[valenumr]

Sure, if every room has a (current) guest in it.

So is "every" even really defined on an infinite set?

 

[Lars Krogh-Stea]

The paradox is that the hotel is full (each of the infinite number of rooms has someone in it), but we can still fit an infinite number of people into their own rooms. To people unfamiliar with transfinite numbers (which is probably the majority of people on earth), this is a paradox.

The "practical" solution is the usual one; namely where each existing guest moves from room N to room 2N. This scheme exhibits the isomorphism that demonstrates that the two sets--the set of positive integers and the set of even positive integers--have the same cardinality.

In the solution you described, it's impossible to say where a given guest has to move, so this scheme doesn't demonstrate the isomorphism between the two sets.

This, and PeroK's answer, is what i was after. I wanted to explore the common "riddle/paradox" vs the uncommon (for normal people like me) paradox that exists within a constrained hypothetical reality shaped to demonstrate a specific point about sets of numbers. My suspicion was that this was not a "riddle" as it was presented on TV in a quiz-like manner (where the point of the show is to deliver obscure knowledge in a humorous way, and no one is expected know the answer). I know wery little about numbers, but was intrigued by the presentation and read the wikipedia article. I do know psychology however, and had a feeling it was somewhat like the Monty Hall problem (but maybe reversed). I've seen people discuss the math involved in that problem for pages, not realizing that it's their biases and bult-in heuristics that keeps them from seeing the answer.

 

[Lars Krogh-Stea]

Did you mean Cant(or)

Cant(or) can't.. panini or punani. It's alle the same ;)

 

[Lars Krogh-Stea]

It is still the case that if there was one person for every real number, this wouldn't work

The Zeno's paradox reveals a deep truth about the convergence of infinite sums, and the hotel is intended to reveal deep truths about infinity. Dismissing it as 100% trivial is only going to keep you from learning things

You are totally right. What i was trying to get across was that these kind of problems/paradoxes lose their aspect of teaching/reflection when they cross over from the experts, like you, to the public, like me. The hotel paradox is full of insights when presented to the academic audience, but is presented like a hard riddle when presented by common people to common people. Furthermore, my point was that, if all it was, was a riddle - then the soulution could be a simpler one. My example was to make them all go out and then enter again. All the academic value is lost in translation anyway. In the same way the Monty Hall problem is often presented like there exists an incomprehencible statistical solution that normal people won't understand, when in reality most people should understand it quite well if they would only let their guard down.

 

[Mark44]

I'm not sure how time comes into play with the thought experiment. I hadn't thought about that previously.

Moving all of the current occupants to a room one number higher will take some finite, nonzero amount of time, so if you have to do this operation for an infinite number of new arrivals, it will take an infinite amount of time. The simpler way, moving each current occupant to the room number that is twice their previous room number, also takes a finite amount of time, but it happens only once.

So is "every" even really defined on an infinite set?

Why not? Tell me a positive integer room number, and I'll tell you whether it has an occupant. Repeat until you get tired.
There is a trivial isomorphism between the set of current occupants and the set of positive integral hotel room numbers. Each person has a room, and each room has a person. That's what the statement "the hotel is full" means.

 

[Mark44]

It's like the paradox where you would never reach the door, because first you need to cover half the distance etc.

That's essentially Zeno's Paradox, except in that one, it was about Achilles shooting an arrow. It was a paradox back when he came up with (500 BC?), but once you understand the concept of infinite series, it's no longer a paradox.

$\frac 1 2 + \frac 1 4 + \dots + \frac 1 {2^n} + \dots$ is a convergent series that sums to 1. This is what Zeno didn't know.

 

[Lars Krogh-Stea]

That's essentially Zeno's Paradox, except in that one, it was about Achilles shooting an arrow. It was a paradox back when he came up with (500 BC?), but once you understand the concept of infinite series, it's no longer a paradox.

$\frac 1 2 + \frac 1 4 + \dots + \frac 1 {2^n} + \dots$ is a convergent series that sums to 1. This is what Zeno didn't know.

Unless you add quantum physics to the mix, then I might observe that, statistically, you've suddenly crossed the treshold :D Okay, I'm not making much sense. It's past bedtime in Norway.. Thanks for the lessons guys :)

 

[Mark44]

Unless you add quantum physics to the mix, then I might observe that, statistically, you've suddenly crossed the treshold :D Okay, I'm not making much sense.

No need for physics here, quantum or otherwise.

 

[valenumr]

Moving all of the current occupants to a room one number higher will take some finite, nonzero amount of time, so if you have to do this operation for an infinite number of new arrivals, it will take an infinite amount of time. The simpler way, moving each current occupant to the room number that is twice their previous room number, also takes a finite amount of time, but it happens only once.

Why not? Tell me a positive integer room number, and I'll tell you whether it has an occupant. Repeat until you get tired.
There is a trivial isomorphism between the set of current occupants and the set of positive integral hotel room numbers. Each person has a room, and each room has a person. That's what the statement "the hotel is full" means.

I still don't see how time applies to the question. Its not really a ln element of the "paradox" as far as I can see.

But to the latter point, I would saying implying "each" to infinite sets is sketchy. It implies the rooms and guests are finitely enumerable. We can't really say infinity == infinity. What's the difference between 3*inf / inf vs. 100*inf / inf?

 

[valenumr]

Moving all of the current occupants to a room one number higher will take some finite, nonzero amount of time, so if you have to do this operation for an infinite number of new arrivals, it will take an infinite amount of time. The simpler way, moving each current occupant to the room number that is twice their previous room number, also takes a finite amount of time, but it happens only once.

Why not? Tell me a positive integer room number, and I'll tell you whether it has an occupant. Repeat until you get tired.
There is a trivial isomorphism between the set of current occupants and the set of positive integral hotel room numbers. Each person has a room, and each room has a person. That's what the statement "the hotel is full" means.

There is a trivial isomorphism of (R>=0, +) and (R, *). How is that relevant?

 

[PeroK]

I'm not sure how time comes into play with the thought experiment. I hadn't thought about that previously.

Let's assume we have an empty hotel with a room for every positive integer and an ordered list of a countably infinite set of guests. If it takes one minute to accommodate each guest, then each guest gets their room sometime. The 60th guest gets their room after an hour; the 1440th guest after one day; and the millionth guest in less than two years.

Everyone on the list knows when they will get their room. Everyone gets a room eventually. Although, of course, there is never a time when everyone has a room!

Alternatively, the hotel has a process where they may accommodate the first guest in a minute, the next guest in 30 seconds, the next in 15 seconds, and they can continue getting faster and faster indefinitely. Now, everyone will be accommodated within 2 minutes.

 

[valenumr]

Let's assume we have an empty hotel with a room for every positive integer and an ordered list of a countably infinite set of guests. If it takes one minute to accommodate each guest, then each guest gets their room sometime. The 60th guest gets their room after an hour; the 1440th guest after one day; and the millionth guest in less than two years.

Everyone on the list knows when they will get their room. Everyone gets a room eventually. Although, of course, there is never a time when everyone has a room!

Alternatively, the hotel has a process where they may accommodate the first guest in a minute, the next guest in 30 seconds, the next in 15 seconds, and they can continue getting faster and faster indefinitely. Now, everyone will be accommodated within 2 minutes.

I guess I've never seen the question formulated with time as a parameter. I always saw it as a comparison of infinities and nothing more. I'm not sure how much it changes things with constant time, I'll have to think about it. But in the immediate, I suppose it could be divergent timewise.