Talk:Infinity

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Revisions

I'm going to make a few revisions to the page:

• The bit about infinity being taken out of physics because the universe is finite doesn't make any sense. Infinite quantities show up all over in physics. Heck, renormalization in QFT is a lot weirder to me than anything infinity-related mathematicians ever do.
• I think the convergence section can be made more transparent. I'll give it a shot.
• Nobody's going to care about the continuum hypothesis unless they already have reason to believe that $\mathbb {R}$ is bigger than $\mathbb {N}$. So I think that if there's going to be a more technical section in this article, Cantor's diagonalization argument makes more sense to use. I'm going to insert this and leave the continuum stuff in place, but I'll delete the more technical part in a couple days if no one objects.
• The only alleged example of a non-archimedean ordered field that's actually defined in the article (affinely extended reals) isn't an ordered field at all. Attempting to clean up.
• The bit about non-archimedean space and time is a bit strange to me. It appears that plenty of people have suggested looking at non-archimedean spacetime, and not just cranks. But I'm not really sure what the page is trying to say -- what do we mean by "non-archimedean" in this setting? Probably no one wants to try to put a physically meaningful ordering on space, and that's the only sense of non-archimedean defined here. I'm inclined to just cut this bit -- it needs to be fleshed out to make sense, it's mostly the author's speculation, and it's not central to understanding infinity.
• The misapplications section is junk. Of course if you want to claim some kind of ergodic hypothesis on the universe and it's unbounded you'll get very unlikely events to happen. That's basically what ergodic means. So that's not really much different from just claiming straight out that "The universe is infinite, so everything that can be imagined must exist somewhere!" I'm putting this back how it was. Maybe there's some argument that this is true, but it had better be more than "infinity! ergodic!".

This is why I hate maths. ^moral 18:03, 27 March 2011 (UTC)

If the universe is infinite

ISn't this kind of a given? IF somthing is actually possible, even given an infinatly small probability, and given infinate time and space then it is a logical conclusion that anything that can happen (IE not violating the laws of physics, mathematics, chemistry etc) will happen?--BenB (talk) 16:54, 1 June 2011 (UTC)

In rational numbers, there's an infinite number of numbers between 2 and 3. But none of them is 4. :) Anyway, I've heard the phrase "finite but unbound" applied to the universe, though I'm not sure if this actually is the current model.--ZooGuard (talk) 17:02, 1 June 2011 (UTC)

(EC) No. Think about a toy model, where the universe consists of a single particle, moving in a straight line. There are configurations that are "possible" insofar as they do not contradict the laws of physics, but that will nonetheless never be reached from our initial condition -- there's no law that a priori prevents the ball from being somewhere off the line, but none of these configurations will ever arise from the conditions we started with, because the ball will keep moving in a line. Similarly, there are configurations of the universe that are physically possible, but we'll never actually reach all of them. (Well, maybe you'd argue that these have probability zero. But then what does it mean to say it "can" happen?)

Probably the right abstract setting for talking about questions like this is the language of wp:dynamical systems. Some systems will have the property you describe: every possible setup will eventually be reached, or close. One kind of system that satisfies this is called "ergodic". But I don't think you'll find a lot of people who would claim that the universe is ergodic (assuming one can even make sense of the statement in the first place). In fact I doubt you'll find anyone except Maratrean, who advocated that view on a version of this very page. --MarkGall (talk) 17:08, 1 June 2011 (UTC)

Mark, how could one decide whether the universe is ergodic or not? I assume one approach is to start with some rough model (since we don't know the exact laws of physics) and ask if that model is ergodic?

In terms of what BenB says, it is true based on the probability theory that if an event has non-zero probability, then over an infinite number of times it will almost surely happen — which means the probability of it ever happening will be one, although things with probability one can still fail to occur. If it has an infinitely small probability (as opposed to an arbitrary small non-infinitesimal probability), I am not sure what will happen. Probability theory is usually defined over the classical reals, which do not permit infinitesimals. One could try to work out probability over some non-classical field (e.g. hyperreals, surreals, etc.) which provides infinitesimals; but I am not sure what the result would be.

From a physical perspective, let me present two arguments. One is based on classical statistical mechanics. The particles of objects with non-zero temperature are vibrating, and the higher the temperature the greater the vibration. However, this vibration is random in terms of the direction the particle moves in, how far it moves in that direction before turning in another, etc., so the microscopic motion responsible for temperature does not cause macroscopic motion. But if the particles are moving in random directions, there is a non-zero probability the particles will all move in the same direction at the same time, which would cause the object to engage in macroscopic motion. This probability is extremely small — although my desk could start levitating right now without violating the laws of physics, the probability of it doing so is so tiny I should not be worried. But, the probability being non-zero, in an infinite universe, the probability of it eventually happening will be one. So (assuming my desk is going to exist forever), one day my desk will levitate, but maybe only after an unimaginable (yet finite) number of years. (After enough years, the probability is very close to 1, even though the amount of years is finite). So, this would seem to be true of any system described by classical statistical mechanics, very many particles moving randomly. Your toy model is obviously not such a system though. But, suppose we want the system to rearrange itself into some arbitrary different configuration - random motion can do that with an unimaginably small non-zero probability, in an infinite universe with probability of one.

My second argument is essentially the same, but rather than relying on classical statistical mechanics, it relies on quantum theory. Suppose I have an electron in a box. I could observe it in many positions in the box - its wave-function gives the probability amplitude the particle will be found at a given place. But the domain of its wave-function is not limited just to the box, it extends beyond it. So, the electron can be found outside the box, although if the box is big enough, the probability is quite small. The electron's wave-function isn't just limited to the box and its immediate surrounds, however, it extends without some abrupt cutoff throughout the entire universe. So the electron could suddenly move from my box on Earth to somewhere in the Andromeda Galaxy, although the probability of it doing so is very low. But, we can get the universe into almost any arrangement of matter, since any particle can "move" anywhere else instantaneously with a certain probability, due to its wave-function extending throughout the entire universe. So, every particle in the Milky Way galaxy could suddenly rearrange itself to form a very different galaxy; the probability of that happening is unimaginably small, but not zero. Given the probability of that event (and its maintenance over time) is non-zero, in an infinite universe it has probability one of eventually occurring.

So this is my two main arguments why I think the real universe (as opposed to a very simple toy universe like the one you describe) is ergodic, as you say. I think almost every universe which is sufficiently large, and which contains sufficient entropy, will be ergodic. Your toy universe won't be, because it it so small (only one particle) and has so little entropy. But I think almost every universe whose particle count and entropy are on the same order as our own will be ergodic. What say you Mark? (((Zack Martin)))^™ 19:55, 1 June 2011 (UTC)

Well, here's another setup to which much the same argument applies. Take a three-dimensional lattice (just a 3d grid) with an ant doing a random walk around its points: it starts at 0 and moves either up, down, left or right, front or back with probability 1/6th each. Now, for any point in the lattice, there's a nonzero probability that it will be reached by the ant eventually, and so if the ant marches for infinite time, it will reach the point with probability 1, right? In fact no! you can compute it -- the probability is less than one (in fact quite small for far-off points). The point is that as it moves, points in the direction opposite the movement get smaller and smaller probabilities, and they get so low that it won't reach almost surely. The probabilities change with the state. (OK, this is with discrete time, but the continuous analog has the same result, just harder math). The same thing applies to your example of the desk, I think. As things happen, the probability of the desk jumping gets smaller and smaller. Eventually your desk will be destroyed, and the probability of it reassembling and then jumping are orders of magnitude smaller still. They decrease so fast with time that it is not probability 1 that it will occur eventually.

In general systems like this won't have recurrent behavior (and what we're talking about is much weaker still than actual ergodicity) -- something special has to happen. If you want physics to back up your story of circular time, you'll actually have to do some specific physical arguments. It's not gong to happen for "general reasons", even if you make all kinds of assumptions.^{— Unsigned, by: MarkGall / talk / contribs}

Mark, can you provide some references for your statements about a lattice random walk? I see, in the lattice case, what you are saying is basically Pólya's random walk constant for d=3 — around 34% probability of return to origin... Polya's constant is specifically for the origin... what applies to the non-origin? Also, if you could supply a reference for the continuous case as well. This is something I find very interesting.

Your argument on random walks may well disprove my argument from classical statistical mechanics, although I'd need to do some more research on that. Also, statistical mechanics is not just a single random walk of a single particle, but the random walk of a large number of particles... Anyway, I don't think it addresses the wave-functional argument though, since that does not involve any random walks. (((Zack Martin)))^™ 09:25, 2 June 2011 (UTC)

Hmmm, apparently actually computing the constants in dimension 3 is harder than I remembered. Certainly the probabilities are less than one and should decrease exponentially as you move further away -- I'll see if I can hunt something down today. The continuous analog of a random walk is a wp:Wiener process which is basically Brownian motion. It also returns with probability 1 in dimensions 1 and 2, but not dimension 3, for basically the same reasons. I'm not sure what the wave-function argument is exactly, but I think it could have a similar problem. My grasp of quantum mechanics isn't so good, but I'd expect that to run into the same problem. When the electron is in the box there's some minimum, nonzero probability that it will escape at any given time -- it's probably the probability at the center of the box. With a probability of at least 1/100000 of jumping out at whatever time, it surely will eventually. But we're asking for something more like the electron jumping into the box from starting outside, and there's no longer a minimum probability. As it moves further away, the electron is less and less likely to jump in, and the probability goes as close to 0 as you want. This is essentially what happens in the random walk -- the probability shrinks very fast as the system evolves, so I don't see why the electron would eventually almost surely end up inside the box.

Probably the same arguments would apply with many particles -- things will just get even more unlikely. Or, if you want to imagine that there's some sort of "phase space" of the universe, with one point corresponding to each configuration, you can think of the time evolution as a random walk of a single particle (corresponding to the current state) on phase space. But I'm just making up all my physics here. --MarkGall (talk) 12:50, 2 June 2011 (UTC)

Thanks for the info about the Wiener process. This is really a topic I need to study in a bit more detail before saying anything further... I have set myself a set of mathematical questions to which I seek answers. Thanks for your valuable input. (((Zack Martin)))^™

My answer to Ben is that just because there is infinite space and time doesn't mean anything that can happen will happen (in an infinite number of times and places). It just means it goes on forever. The math here is serious, and I don't know it, but saying "vanishingly small times infinity equals one" is probably incorrect. ħuman 02:24, 2 June 2011 (UTC)

Maybe the above example Maratrean and I were discussing is also a nice one for Ben. Imagine an infinite hotel made out of cube-shaped rooms all stacked together. A person in one of the rooms can go into any of his six neighbors -- up, down, left, right, front, back. Suppose the guy was visiting a friend in the room next to his, but he forgets how to get back to his own room. He could start doing a "random walk", where he rolls a die and goes into the one of the six neighboring rooms depending on what the die comes up as. He keeps repeating this random thing until he gets back to his room. It's certainly possible (there's a nonzero probability) that this guy will eventually find his room. But is it certain that if he goes long enough, he actually will? No -- if you do some pretty tricky calculation, you find that his chances of making it home, ever, are only about one in three! --MarkGall (talk) 13:43, 2 June 2011 (UTC)

But, to make this more accurate as a model of the universe, we'd need to add some more elements:

1. Not just a single particle in a random walk, but many. If space is infinite, arguably the particle count should be infinite too; we could make our initial conditions that every Nth room is occupied. Alternatively, if space is finite and bounded (a toroidal lattice), then we could have a finite particle count.
2. Only one particle can occupy a room at a time, so some change to the movement condition to prevent this. (e.g. still one-sixth chance in each direction, but if you can't go there, stay where you are)
3. Particles are fungible, so we don't actually care about the history of any individual particles, we care about configurations/arrangements of particles. We are interested in the probability of a given configuration recurring, regardless of whether it involves the same particles playing the same roles.
4. Furthermore, we should count a configuration reoccuring if a translation, rotation, or reflection, of the configuration occurs. This is because, if we are inside the configuration, we don't care about how the configuration relates to the rest of the universe.
5. We could look at either (1) does the entire universal configuration ever recur, or (2) does some substantial subconfiguration recur?

Now, with this more complex model, does a result similar to Polya's still hold? Maybe it does, maybe it doesn't, I don't know. But if a Polya-style result doesn't hold here, then we have a model which justifies my claim that the universe will repeat. And, will every possible eventually configuration occur, or only some? That would answer the question of whether "anything that could possibly happen eventually will". (((Zack Martin)))^™ 23:51, 3 June 2011 (UTC)

But that's not even interesting, or on topic. ħuman 06:12, 4 June 2011 (UTC)

You may not find it interesting, but I find it interesting. And it is definitely on topic; Mark Gall's simple toy universe model is definitely not ergodic, but it is an open question whether or not some more complicated universe models might be ergodic? (((Zack Martin)))^™ 10:57, 6 June 2011 (UTC)

1/0

Can we add a section about 1/0 onto the page? I'm not familiar with the <.math> tags so I don't know how to format it. However, it is worth noting that 1/0 is considered infinity. Consider:

1 / X

1 / .1 = 10

1 / .01 = 100

1 / .001 = 1000

....

1 / .0000000001 = 1E10

Given this, what would a number look like as X approached zero? It would get incredibly large. Unfathomably large. Infinitely large. Anim (Carfa) 12:50, 5 April 2017 (UTC)

ZFC

The article uses "ZFC" without ever first explaining what the heck it stands for. This seems unfair to readers. Vivisectionist (talk) 16:00, 12 December 2020 (UTC)

Borked

It seems something has broken the crap out of this page. I dont know Wikicoding enough to even start. Revolverman (talk) 18:52, 28 August 2023 (UTC)

It's an issue with <math> that's been broken since the last MediaWiki update. Plutocow (talk) 19:47, 28 August 2023 (UTC)