5 ms·

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:

by mojuba 17y 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.

byrneseyeview 17y 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?
- mojuba 17y 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.
  - byrneseyeview 17y ago
    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. There is a pretty good Overcoming Bias post on why this is wrong: http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/ http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/
    - mojuba 17y ago
      Nice read, thank you. So he understands that when we lose sight of that photon, from that point on our knowledge about it becomes a hypothesis based on laws of physics. Unfortunately laws of physics are neither 100% accurate nor complete - there may be some laws not discovered and formulated yet and even those discovered never give you 100% accuracy in making predictions. However, we are talking about infinities, which is a different kind of beast. My main thesis is that in order to measure infinite distance you need infinite time, which we simply don't have at our disposal, and thus infinity in any of its form can't be observed in principle.
mnemonicsloth 17y 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.
- mojuba 17y 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.
  - slackenerny 17y ago
    This is akin to entering room room full of engineers and shouting "I can fly! Prove I can't!", what do you mean, prove you can. Take any serious book on modeling bus and routing infrastructure (and not vocational guide on using protocols designed by someone else) and you will see plenty of partial differential equations.
    - mojuba 17y ago
      Just one example please. Ideally, a theory based on which engineers built something, but not the other way around, i.e. when something exists and some theorists are trying to build a de facto model of it. Edit: a theory that involves infinities. Edit: I'd like to warn you though, that because I'm more or less familiar with the subject, I think it will be very difficult to find what I'm asking for.
      - jgrahamc 17y ago
jerf 17y 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.)
- mojuba 17y ago
  As much a joke as the original paradox. I think it's a joke in a very broad sense.
jgrahamc 17y 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?
- mojuba 17y 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.
  - jgrahamc 17y ago
    Can you explain the difference between i and inf? What do you see as the difference in existential significance?
    - mojuba 17y ago
      Among many other things, there is a question whether space and time are descrete or continuous, or in other words, whether 1/∞ particularly exists or not. It has implications even in understanding how intelligence works, that is, whether brain is a discrete machine or not. I don't know if i has such broad and important implications.

by mojuba 17y ago

byrneseyeview 17y 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?

mojuba 17y 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.
- byrneseyeview 17y ago
  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. There is a pretty good Overcoming Bias post on why this is wrong: http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/ http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/
  - mojuba 17y ago
    Nice read, thank you. So he understands that when we lose sight of that photon, from that point on our knowledge about it becomes a hypothesis based on laws of physics. Unfortunately laws of physics are neither 100% accurate nor complete - there may be some laws not discovered and formulated yet and even those discovered never give you 100% accuracy in making predictions. However, we are talking about infinities, which is a different kind of beast. My main thesis is that in order to measure infinite distance you need infinite time, which we simply don't have at our disposal, and thus infinity in any of its form can't be observed in principle.

mnemonicsloth 17y 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.

mojuba 17y 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.
- slackenerny 17y ago
  This is akin to entering room room full of engineers and shouting "I can fly! Prove I can't!", what do you mean, prove you can. Take any serious book on modeling bus and routing infrastructure (and not vocational guide on using protocols designed by someone else) and you will see plenty of partial differential equations.
  - mojuba 17y ago
    Just one example please. Ideally, a theory based on which engineers built something, but not the other way around, i.e. when something exists and some theorists are trying to build a de facto model of it. Edit: a theory that involves infinities. Edit: I'd like to warn you though, that because I'm more or less familiar with the subject, I think it will be very difficult to find what I'm asking for.
    - jgrahamc 17y ago

jerf 17y 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.)

mojuba 17y ago
As much a joke as the original paradox. I think it's a joke in a very broad sense.

jgrahamc 17y 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?

mojuba 17y 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.
- jgrahamc 17y ago
  Can you explain the difference between i and inf? What do you see as the difference in existential significance?
  - mojuba 17y ago
    Among many other things, there is a question whether space and time are descrete or continuous, or in other words, whether 1/∞ particularly exists or not. It has implications even in understanding how intelligence works, that is, whether brain is a discrete machine or not. I don't know if i has such broad and important implications.