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Richard87
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Could an imaginary hotel with an infinite number of rooms be packed?
the hotel of interest has infinitely many rooms, so if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1. By repeating this procedure, it is possible to make room for a countably infinite number of new clients: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room N to room 2N, and all the odd-numbered rooms will be free for the new guests.
Richard87 said:Apparantly an hotel with an infinite amount of rooms could never be packed, even with an infinite number of guests.
Evolver said:This is a flawed argument because what the concept "infinite" actually equates to in mathematics/physics is "undefined". For example, in physics it is said that in order to make matter attain the speed of light it must have infinite energy... which comes out in equations as undefined. In black holes it is said that gravity compresses matter to such a condensed state that it has infinite density... which comes out to undefined in equations. And that is no less true than the question you have proposed. You have not defined how many rooms must be filled therefore there can be no answer.
SW VandeCarr said:Infinity is in fact a valid concept subject to rigorous treatment in set theory. Even in physics cosmologists are considering the possibility of an infinite universe. The hotel example is an old one often used in teaching students about infinite sets.
In terms of my argument: 'The hotel is full but we can always make room for more.': you might ask if every room has a guest (one to one correspondence) how can there be a vacant room for a new guest? There is an answer. Can you figure it out?
Or, y'know, integers. The natural number 1 is no less abstract than anything else you listed there -- its only distinction that it, and some applications, were taught to you at a very young age.Evolver said:When scientists use infinity as a concept they do so in the sense that they also use virtual particles or imaginary numbers.
Hurkyl said:Or, y'know, integers. The natural number 1 is no less abstract than anything else you listed there -- its only distinction that it, and some applications, were taught to you at a very young age.
Evolver said:Very true, that is a good point, Hurkyl. It makes my point about infinity being undefined even more true, because 1 is essentially no different than infinity to the universe, these are all man made concepts and as such only produce man made results. So the whole hotel question is already biased based on the idea that the universe would never produce such a paradox, only man can abstract this concept from exaggerated observations of the universe itself.
SW VandeCarr said:Present observations seem to indicate the universe is flat which raises the real possibility that it is Euclidean and infinite. I started a topic recently in the Cosmology Forum entitled "Questionable article in Scientific American" and linked to a summary of an article by physicist Max Tegmark regarding an infinite universe. It so outlandish that sounded frankly cranky to me. I was informed that though it is extremely weird it is not science fiction, but reasonable scientific speculation and not in violation of the laws of physics as they are currently known.
Evolver said:Well the problem with speculation is that it completely rides on what theory you subscribe to. According to Relativity, the universe is constantly stretched and distorted. According to String Theory it has upwards of 9 spatial dimensions and a time dimension. M-theory indicates that it is the result of two interacting, higher dimensional branes.
Once you start dealing with alternate dimensions, ideas of infinity in a finite space arise, much like the Klein Bottle. At our current limited state of understanding, it is nothing more than speculation, though I do think it is an interesting idea.
SW VandeCarr said:Very fast response. I just wanted you to know I added one more sentence just after first posting: The principle objections (to an infinite universe) are philosophical, not scientific.
SW VandeCarr said:Very fast response. I just wanted you to know I added one more sentence just after first posting: The principle objections (to an infinite universe) are philosophical, not scientific.
DocZaius said:Could you elaborate on what types of philosophical objections there would be to an infinite universe, aside from "I don't like it that way?" Not saying you have such objections, just curious about what's being said.
Evolver said:Yes. The universe has never defined any infinite quality of itself in any way that we are yet aware of. So in actuality, assuming the universe could be infinite is the actual use of philosophy, not the other way around.
DocZaius said:Could you elaborate on what types of philosophical objections there would be to an infinite universe, aside from "I don't like it that way?" Not saying you have such objections, just curious about what's being said.
DocZaius said:I think assuming either way is philosophizing. On matters untestable by science, isn't it best to say "I don't know?" That is why I asked my original question. I can't think of any philosophical objection (on either side of the issue) that would hold any weight.
Evolver said:In matter's unsure to science usually Occam's Razor is invoked, saying that if faced with many possibilities that lead to the same answer that the least complex should be taken until proven otherwise. It prevents unnecessary assumptions and cuts out non-important factors.
In this case, saying that the universe has never defined any infinite quantity thus far, so it's safe to assume it's not infinite, would be invoking Occam's Razor. Speculating that it's infinite (though always possible) is philosophy at this current time.
DocZaius said:Under what conditions would it be possible for the universe to even be able to define an infinite quantity?
From my viewpoint it seems that Occam's Razor would invoke an interpretation of infinity, at the very least on the micro scale. There is no reason to believe that the fabric on the microscale isn't continuous. And if your objection is Planck length, my understanding (although it could be wrong) of Planck length is that despite its use in physics, it does not physically signify a "smallest possible distance"; which would imply that the universe is made of discrete parts.
SW VandeCarr said:Infinity is in fact a valid concept subject to rigorous treatment in set theory.
Ack, I can't believe I missed this. What could you possibly mean by that assertion? Or are you doing something weird like confusing "well-defined number" with "number whose decimal expansion has finitely many nonzero digits"?debra said:I find Pi is very interesting in that it appears never to be fully defined. There is a philosophical reason that must be so.
The concept of infinity is quite well-defined.Evolver said:This is a flawed argument because what the concept "infinite" actually equates to in mathematics/physics is "undefined".
Pi is also very well-defined. For example, see http://en.wikipedia.org/wiki/List_of_formulas_involving_π. Just because there remain some open questions about pi (e.g., whether pi is normal) does not mean pi is not well-defined.debra said:I find Pi is very interesting in that it appears never to be fully defined. There is a philosophical reason that must be so.
DocZaius said:There is no reason to believe that the fabric on the microscale isn't continuous. And if your objection is Planck length, my understanding (although it could be wrong) of Planck length is that despite its use in physics, it does not physically signify a "smallest possible distance"; which would imply that the universe is made of discrete parts.
D H said:The concept of infinity is quite well-defined.Pi is also very well-defined. For example, see http://en.wikipedia.org/wiki/List_of_formulas_involving_π. Just because there remain some open questions about pi (e.g., whether pi is normal) does not mean pi is not well-defined.
Hurkyl said:Ack, I can't believe I missed this. What could you possibly mean by that assertion? Or are you doing something weird like confusing "well-defined number" with "number whose decimal expansion has finitely many nonzero digits"?
debra said:What I am trying to say is that a curve cannot be a true representation where space is discrete can it? At a micro micro level there is no 'curve' at all. Think of it as pixelated if you want an easy mind picture. So there can in reality be no such entity as a curve. It can only be curved to so many degrees of precision. Again, at heart, its an infinity problem yet again isn't it? Is space continuous - I say no! There is a philosophical problem with a so-called curve - it is better thought of as a summation of steps.
debra said:A rigorous treatment in set theory? I am not impressed. Should I be over-awed by that statement?
DocZaius said:So your argument is that for spacetime to be continuous, that would mean that infinity would be involved. And therefore it can't be. Why can't infinity be involved at scales we cannot measure? Why is that a logical impossibility? As far as I know the quantization of spacetime is still a postulate and is actively researched. Also, the calculus summation of infinitesmall intervals is a mathematical tool. It does not prove or disprove the physicality of a discrete spacetime.
WaveJumper said:This discussion is going in the wrong direction. It's not that there is no infinity in Nature, quite the opposite. The question is why we see discrete objects that we label with integers?
Everything in nature is in fact infinite, the wavefunction of every single object in the universe is soaked in around it to infinity(consequence of SE). The real question is why we see single laptops, single cats, single everything, which is the heart of the measurement problem. We are inferring that "objects" are not infinite based on how we perceive them, not how they truly are, apart from our perception of them. Our perception is incredibly flawed and at odds with most everything that has come out of physics in the last 100 years.
The only way around 'infinity-discreteness' is to solve the MP. The solution may turn everything we have assumed so far upside down.
WaveJumper said:This discussion is going in the wrong direction.
SW VandeCarr said:It certainly is. To begin with, the OP was a about an imaginary hotel: just in case some people didn't understand that. The discussion of real physical infinity is off topic and in any case, not an issue that can be addressed by current science.
The subject was infinite sets, a purely mathematical concept. If you want to argue that a mathematical argument is flawed, then you have to argue in terms of a mathematical context. Even grade school children that I teach as a volunteer understand that.
Perhaps some people thought an infinite hotel was real. Well if was, it could blow out the competition by charging really low prices.
No, a hotel with an infinite number of rooms can never be full. This is because infinity is a concept that represents something without an endpoint, so there will always be an available room for someone to stay in.
In mathematics, infinity is a concept that represents a quantity without an endpoint. In this scenario, it means that the hotel has an unlimited number of rooms, with no limit to how many rooms can be added.
Yes, a hotel with infinite rooms would be able to accommodate an infinite number of guests. This is because infinity has no limit, so there will always be enough rooms for all the guests.
In an infinite room scenario, the hotel could use a mathematical approach to assign rooms to guests. For example, they could use a system of prime numbers to assign rooms, where each prime number represents a unique room.
While a hotel with infinite rooms may seem like a hypothetical scenario, it could have practical applications in the world of mathematics and theoretical physics. It could also be used as a thought experiment to explore the concept of infinity and its implications.