Is Infinity Real?

nwr writes:

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The other way is to see it as saying that movement requires infinitely many stages, and we can only carry out a finite number of those.

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And that would be a false premise. Each stage takes less and less time such that you can carry out an infinite number of them in a finite time.The original problem was brought up as a means to point out how the way in which you count something can have an effect on the cardinality of the set which you calculate, but applies here:For this thought experiment, we must assume a few things:1) Superman and Captain Marvel both exist and can move any amount of speed in any amount of time.2) There are an infinite number of coconuts in a pile, all numbered: 1, 2, 3, ...3) There is a pit that can hold them all.At noon, they decide to play a game. Captain Marvel throws coconuts numbered 1 and 2 into the pit. Superman flies in, grabs coconut number 1, and tosses it out of the pit.They then sit around for half an hour, discussing their various exploits, when at 12:30, Captain Marvel throws in coconuts 3 and 4. Superman flies in, grabs coconut number 2, and tosses it out of the pit.They repeat the process, only waiting 15 minutes this time. In go 5 and 6, out comes 3. This process repeats, each time halving the amount of time they wait between tossing coconuts.When 1 pm comes around, and 1 pm always comes around, how many coconuts are in the pit?The answer, of course, is none of them. They have all been tossed out. Why? Because for every coconut you can identify, I can give you a precise time when it was tossed out of the pit. Number 1 came out at noon, number 2 at 12:30, number 3 at 12:45, etc., etc.And thus, an infinite process can be carried out in a finite amount of time.What Xeno had wrong is that yes, you have to cover an infinite number of half-distances in order to reach the end, but each of those half-distances takes half as much time to carry out.If Xeno had had the tools of analysis (not so much calculus, per se), he would have been able to see that his infinite sum converged.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

Dr Adequate writes:

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Think of time as being like space, and imagine viewing it sub specie aeternitatis --- a God's-eye view if you will. From that point of view, wouldn't it look even odder if it just started at some point?

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No, not really. If there were an infinite past, then processes that would reach equilibrium after an infinite amount of time (or even a sufficiently large amount of time) would have done just that. Since they haven't, then there must not be an infinite past.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

cavediver responds to me:

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If there were an infinite past, then processes that would reach equilibrium after an infinite amount of time (or even a sufficiently large amount of time) would have done just that. Since they haven't, then there must not be an infinite past.

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Perhaps you could list these processes so that we can check the veracity of your claim?

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The heat death of the universe, for one.Now, one might suppose that there is a way for the universe to "restart" itself, cycles of Big Bangs and such. But that would require a significant change in the Second Law regarding the nature of energy. Is it possible to have an infinitely recurring process whereby enough energy to populate the entire universe is generated? We have yet to find any perfect system anywhere else...why should the universe be an exception?At any rate, this is an interesting concept. It would mean that while the observable universe in its current state "started" at a finite point in the past, there is an unobservable universe. This, of course, raises the question: If it is unobservable, does it even exist? If we thought that string theory had a hard time due to the inability to run an experiment that tested it, how much more difficult would it be to run tests on a universe that by its very definition is incapable of being observed? The mathematics may be very beautiful (and I say that as a mathematician who does think that inifnity really exists), but unless we can find some way to demonstrate it, it will be nothing but a pretty framework with no substance.Now, I certainly like the idea. There is something personally disconcerting about a "one-shot" system on the scale of the universe. I don't have the common "logical" problem that comes up of, "If the universe is infinite, how did we get to 'now'?" To my mind, every stretch upon an infinite process is just as real as any other and we can exist anywhere along it, so why not where we are? There's nothing special about where we are, we just are.But again, until we can find some way of examining such a construct, it is nothing but a pretty picture we've painted for ourselves. Might be true, might not be.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

cavediver responds to me:

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but de-Sitter, which is past-infinite, does not.

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But a de Sitter universe has no matter. While it certainly seems like we are becoming a de Sitter-style universe due to the expansion of the universe and the dominance of the cosmological constant, we aren't there now.But that said, if the universe were infinitely old, why haven't we achieved de Sitter status yet? The universe is expanding...and accelerating at that. Unless we are going to suggest some process that kicks out instantons....

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Sure, in a spatially infinite universe.

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So what happened to the First Law? Everything's gotta be somewhere. If there's an infinite amount of energy to be had, where is it?

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I'm not sure about "perfect system"

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Classic Second Law. There is no perfect system. All processes bleed energy that can never be recovered. So unless there is an infinite amount of energy, this process of kick-starting universes over and over again will eventually fail.

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but this is in danger of the fallacy of composition - which is relevant in this topic given the tired old argument of "there's nothing in the Universe that is infinite, therefor the Universe cannot be infinite."

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Except I've already stated that I do think that infinity does exist.

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Which is why we are all about the evidence...

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It certainly is an interesting time to be a cosmologist.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

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Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

cavediver responds to me:

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I'm not sure what you mean by "de-Sitter status".

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A universe which, for all intents and purposes, is devoid of matter.

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Yes, under certain behaviours of dark energy, the Universe will approximate de-Sitter behaviour, but that is not the point.

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Incorrect. That is precisely the point: The universe is expanding...and accelerating at that. Thus, if things keep going the way they are, the universe will be a pretty good approximation of a de Sitter universe: No matter. There will come a point where the only observable universe we will be able to see is the Local Group since we are gravitationally bound. Everything else will have been lost to the expansion of the universe.If there were an infinite amount of time, why hasn't it happened yet?Again, that goes back to the concept of a process that somehow manages to kickstart a new instanton. A perfect process that never fails, never loses energy.

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If there's an infinite amount of energy to be had, where is it?

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Distributed over a spatially infinite Universe.

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You're misunderstanding my question. I'm referring to this posited process that can somehow create an instanton with enough energy to create a universe. With an infinite amount of time, that requires an infinite amount of energy. But the First Law says that everything has to be somewhere. So where is it? If it is infinitely distributed, how did it manage to get here from everywhere else?