Could an imaginary hotel with an infinite number of rooms be packed?
Only if there is an uncountable infinity of guests. Maybe if the Real Numbers decide to hold a convention.
Apparantly an hotel with an infinite amount of rooms could never be packed, even with an infinite number of guests.
Assuming one guest to a room, there can be a one-to one correspondence between guests and rooms. So the hotel would be "full" but they could always make room for more because [tex]\infty+1=\infty[/tex]
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.
The idea of infinity is fine - a number that never ends, but its practical realization cannot be realized because it would imply an infinite information packet size - which cannot exist in reality. So, following on from that concept, the 1/r equation for increasing field strengths must cut-off at some level. There cannot be infinite field strengths for the same reason.
I find Pi is very interesting in that it appears never to be fully defined. There is a philosophical reason that must be so.
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?
Infinity is still undefined by nature though. If something were truly infinite it would have no reason to define itself because doing so would impose a physical limitation. When scientists use infinity as a concept they do so in the sense that they also use virtual particles or imaginary numbers. There is no physical version of them, they simply represent a concept that cannot be otherwise described. Hawking's black body radiation uses virtual pairs at the event horizon of a black hole... it is a concept to explain that phenomenon.
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.
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.
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 is so outlandish that it 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. The principle objections to an infinite universe are philosophical, not scientific.
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.
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.
I totally agree, and personally I'm not against the idea. I just know that, to this date that we are aware of, the universe has never defined anything as being truly infinite (because by definition if you defined it it could not be infinite.)
I see infinite as more of a man-made concept as opposed to a property of the universe... though I could be completely wrong.
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.
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.
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.
One of the consequences of an infinite universe is that's its populated by an infinite number of people just like us. Indeed, they can be identical copies of us. Each one of us has an infinite number of copies. What does that say about the autonomy of the individual? If you are an identical copy of another individual, not just a biological twin, but in every detail of your life including your history. memories parents, ect; then why isn't that person you? How are you distinguished from the other person? Does everyone's life repeat over and over again in some distant place?
If you think what I just said is whacko, I agree, but it is a consequence of infinity. In an infinite universe anything that can happen will happen, and will happen over and over again an infinite number of times!
Not being a philosopher. I don't know if this a valid objection, but I know that significant parts of philosophy as a discipline are devoted to the nature of the individual and the nature of existence.
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.
Under what conditions would it be possible for the universe to even be able to define an infinite quantity that we could satisfactorily interpret?
From my viewpoint it seems that Occam's Razor would imply 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.
My observation of what seems to be continuity everywhere (and that is enough for OR) suggests infinitesmall intervals and I invoke Occam's Razor in that sense.
My point being that I don't think OR favors either interpretation.
Well that's my point. Infinity is a concept created by man to represent a thought he cannot comprehend. But it doesn't mean that it is a truth of the universe. The universe doesn't care about infinity as much as man does ;) I think it's possible we are thinking about it in the wrong way.
Here is another post of mine that deals with the universe as something completely different than what we intuitively think: https://www.physicsforums.com/showthread.php?t=368284
A rigorous treatment in set theory? I am not impressed. Should I be over-awed by that statement?
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"?
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.
Then what do you do with the force is proportional to 1/r problem? Implying infinite forces.
Infinities problems dog physics - they have no answer. Or, lets introduce a cut-off because that is what experiment shows. To define small separations needs massive data chunks.
These data chunks - called metrics in physics - would need to go to infinity if space were continous. Sorry - space must be discrete at the micro level.
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