Hilbert's paradox of the Grand Hotel

http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

by tokenadult • 17 years ago

14 37 17 years ago

en.wikipedia.org

A lot of mathematicians I've known feel a kind of awe for Hilbert that they don't feel even for history-makers like Euler or Gauss. I think little stories like this one are part of the reason why. Hilbert always makes it look easy, even when his results are weirdly counter-intuitive.

Technically, though, there aren't any paradoxes here. It's for the purpose of avoiding paradox that we're willing to talk about infinities in the first place.

slackenerny5• 17 years ago

Technically, though, there aren't any paradoxes here.

The real reason why it is a paradox doesn't have much to do with notion of infinity per se, but with problem of infinite decision aka Supertask.

Which reminds me of an article on physics of supertasks that among other things somewhat outlines how classical theory is more paradoxical before than after quantization http://th-www.if.uj.edu.pl/acta/vol36/pdf/v36p2887.pdf (43pp but easy reading).

Luc• 17 years ago

I really wanted to like this PDF, but the grammar is at times so bad it is often quite hard to parse.

mojuba12• 17 years ago

When you ask client N to move to N+1, what happens is this client becomes temporarily "homeless", or not occupying any room for the duration of this procedure: packing, moving out, asking the next client to do the same and waiting until the next client packs, moves out, etc. So once you want to make room for your new client and start the procedure, one of the existing clients at any given time will be in the lobby, not occupying any room.

Accommodating infitine number of newly arrived guests thus means having infinite number of poor "homeless" guys in the lobby.

So? So... Grand Hotel can accomodate exactly infinite number of guests, but not more than that. If it's full, it's full.

Let alone the fact that infinity doesn't exist because we can't observe it.

byrneseyeview2• 17 years ago

If each person waits until the next room is unoccupied, you only have one homeless person at a time, but it takes infinitely long to add a new guest. However, I don't think that's Hilbert's point. If it's easier, assume that no guests have possession, and that every room has a door to the previous and next rooms. So everyone can get up and go to the next room at the same time.

I always have the sneaking suspicion that these things only feel paradoxical because we have a better intuitive understanding of the "First" room than the "Last" room, and yet we also understand the idea of the "Next" room. In every case, we have a chain of "Nexts" taking us from the concrete "First" to the weirdness-factory "Last".

Let alone the fact that infinity doesn't exist because we can't observe it.

Does someone standing silently behind you cease to exist until you turn around?

mojuba12• 17 years ago

...it takes infinitely long to add a new guest

No, because once the first client moves, the first room is free for the new guest, even though the whole process of moving hasn't finihsed yet, and in fact will never be finished. You will end up having infinite "waves" of moving clients.

assume that no guests have possession, and that every room has a door to the previous and next rooms

My point is, there are intermediate states when you can't strictly say the hotel is full and that every occupant has a room, i.e. one-to-one relation. In your scenario, if you start from the first room, you will always have one room with two occupants, which is against the rules, or in my scenario there will always be one homeless dude.

I understand this is an abstract situation anyway, which can not be experimentally reproduced (for one, it would take infinite time to build this hotel and we'll never see it finished), which leads to the next point:

Does someone standing silently behind you cease to exist until you turn around?

You can never be sure about anything unless you observe it using your own senses and possibly some equipment in the middle. (And it gets even worse in the quantum world, but let's forget that for a moment.) So strictly speaking, when you don't see the person behind you, you can only make a prediction/hypothesis that if you turn around you will see that person again. It's based on some laws of physics suggesting that a person can't disappear, say, in one nanosecond. But hypothesis is just a hypothesis, and it's not observation of reaality, e.g. a fact.

Back to infinities. In the physical world all kinds of infinite values are defined through either infinite distance or infinite time, including both infitely big or small, i.e. ∞ and 1/∞. Space and time are closely related though, because in order to measure a distance you need "some" time, and obviously, to measure or observe infinite distance you need infinite time. Which is not possible for us, humans and therefore infinity is just a hypothesis, and I'm afraid will always be. (Always? You mean, for infinitely long time? :)

Curiously, for some reason I started thinking about infinities this morning and here you go, a post at YCNews.

mnemonicsloth6• 17 years ago

infinity doesn't exist because we can't observe it

Presumably, you are accessing this website using a some kind of personal computer and some communications infrastructure. The engineers who designed these systems all made frequent use of the notion of infinity, particularly in their use of limits and calculus, when these systems were designed.

I am sure that those engineers would love to hear about the equivalent (but necessarily much more complex), infinity-free versions of calculus and limit processes that you have obviously developed. Since infinity is the only simplifying assumption the average engineer is ever forced to make, I'm sure that as a group, they will be very eager to take your advice, even if it means they have to do twice as much work.

mojuba12• 17 years ago

Care to show an example how the notion of infinity can make a programmer's job easier? Particularly in the field of Internet protocols and web applications.

amalcon• 17 years ago

Infinity doesn't "exist" because the notion of "existence" doesn't apply to mathematical constructs. We don't find them in nature; we use them to describe things we find in nature. The number two doesn't "exist", though there are plenty of things it models well.

There are, of course, things that are described by infinity: the energy required to accelerate a mass to the speed of light, the number of points at which a magnetic field can travel through a surface, etc.

jerf• 17 years ago

You're using your physical intuition on a hotel with infinite extent.

Do you see where the problem might be?

(If this is some sort of attempt at a joke, it's not very good.)

mojuba12• 17 years ago

As much a joke as the original paradox. I think it's a joke in a very broad sense.

jgrahamc4• 17 years ago

"Let alone the fact that infinity doesn't exist because we can't observe it."

Do you feel the same way about the complex numbers?

mojuba12• 17 years ago

What do I feel about the square root of -1? I love it. I don't think though it has the same existential significance as infinity. I may be wrong though.

dxjones• 17 years ago

What is missing from this discussion of infinity is the notion that there are multiple infinities that are different from each other.

The integers (like the hotel rooms) are countably infinite. There are only a FINITE number of integers (or hotel rooms) between any two integers (rooms), like between rooms 1 and 100.

The set of real numbers is a larger set than the integers. Both are infinitely large, but the real numbers are a larger infinity. Between any two real numbers, (even 0 and 1), there is still an INFINITE number of real numbers in between.

The notion of a hierarchy of infinities initially seems paradoxical, but once you know how to distinguish countably infinite from uncountably infinite, you have the key concept.

mojuba12• 17 years ago

So what does it explain in this context?

drbaskin• 17 years ago

The reals and the integers do have different cardinalities, but not for the reason you imply. In fact, the rational numbers have the same cardinality (the same infinity) as the integers, but they have the same property that you use to characterize the reals, i.e., that between any two distinct rationals there is an infinite number of rationals.

One way to see that there are as many integers as there are rational numbers is just to find a way to count the rational numbers. There are a number of ways to do this. There is a very nice and explicit way to do enumerate the positive rationals that involves the prime factorization of the integers. Suppose first that p is a prime number. We identify p^k with the integer p^{2k}. For the rational number 1/p^k (p is still prime), we label it by the integer p^{2k-1}. For a rational number p/q in lowest terms, we take the prime factorization of p and the prime factorization of q, apply the above identification to each factor p_i ^k and then take the product. This gives a very explicit bijection between the positive rationals and the positive integers.with

My favorite way, though, is just to make a grid of pairs of integers (numerator, denominator), and then count them by spiraling outward on the grid. This yields duplicates, but that's ok.

slackenerny5• 17 years ago

My favorite way, though, is just to make a grid of pairs of integers (numerator, denominator), and then count them by spiraling outward on the grid. This yields duplicates, but that's ok.

Calkin and Wilf cleverly avoid duplications by arranging reals not on a lattice but on a (Stern-Borcot) tree. http://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree http://www.mathlesstraveled.com/?p=94

mnemonicsloth6• 17 years ago

My favorite way, though, is just to make a grid of pairs of integers (numerator, denominator), and then count them by spiraling outward on the grid. This yields duplicates, but that's ok.

Or just enumerate the naturals in base 11. Denote 9+1 by "/". Interpret "/" as a negative sign when it's the first or last base-10 digit, otherwise as division. Pick an associativity.

jpwagner• 17 years ago

What is the point of the cigar section of the article?

tokenadult• 17 years ago

What is the point of the cigar section of the article?

To discuss a necessary step in mathematical induction, namely the basis.

http://en.wikipedia.org/wiki/Mathematical_induction

Hilbert's paradox of the Grand Hotel

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