Talk:Infinity

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I'll take a crack at expanding this over the weekend. - User \pi 22:20, 8 April 2009 (EDT)

I renew the promise, this is the very next thing I will work on. - π 02:07, 29 June 2009 (UTC)

What about adding "misconceptions of infinity"? The stuff you reverted could go well under that header and it'd be more RW-like than just trying to get an accurate definition, which we usually go to wikipedia for anyway. I mean, infinity is a complicated thing in maths so oversimplifications and people throwing the term around without grasping what it means could be interesting to look at. Scarlet A.pngsshole 02:13, 29 June 2009 (UTC)

That is not a bad idea. My favourite rule of thumb, at least in calculus, is that if you can't rewrite the sentence without using the word infinity you are using it wrong. - π 02:27, 29 June 2009 (UTC)
The divide by 0 thing is true though. I remember talking about it with someone in AP Calc back in high school. The graph lines in a hyperbola go on forever, or until the calculator can't handle anymore. That x=0 limit is indeed negative and positive infaggotry infinity. Candlewick 02:17, 29 June 2009 (UTC)
There are ways of finding integrals of undefined objects, have a look at Gabriel's Horn. - π 02:24, 29 June 2009 (UTC)
The one I know is "mathematically undefined" it's not as simple as "infinity" because you first need to define which infinity (some of which are bigger than others, although still infinite). When you talk about infinity you're either talking about a process - which is the commonly accepted, albiet slightly out of date, way of dealing with it - or possibly cardinality which is numbers like Aleph-zero. Scarlet A.pngsshole 02:20, 29 June 2009 (UTC)
I think we should have an short, snarky but above all accurate article on infinity and its many associated concepts. Cardinality and limits are probably the most important two. Also we could have some fun with the Hilbert hotel example. - π 02:24, 29 June 2009 (UTC)
But how can you describe the Hilbert hotel better than it is in Science of Discworld? That's a great coverage of the subject. Scarlet A.pngsshole 02:33, 29 June 2009 (UTC)

Universe[edit]

Isn't there a slight discrpency on the actual universe and the observable universe. I'm not as hot on cutting edge cosmology. Scarlet A.pngsshole 18:03, 17 August 2009 (UTC)

I would guess that the "observable universe" is finite by definition. My impression was that the rest of it was finite as well, but I'm also no astrophysicist.--BobNot Jim 18:09, 17 August 2009 (UTC)
(from interested inorance) Isn't it finite in 3 dimensions (you'll supposedly end up back where you started if you go far enough in a straight line), infinite in 4 dimensions (spacetime) and absolutely mad in any other number of dimensions? I am eating Toast& honeychat 18:19, 17 August 2009 (UTC)
The universe is a big, flat, finite things, which is a bit lumpy in places. That help? - π 13:26, 7 September 2009 (UTC)
Depends what you mean by finite. Unless my geometry's failing me, I don't think flatness and finiteness are really compatible. Doesn't "finite" mean that the (spacetime) manifold is closed (or at least compact), and flat mean that it's non-compact? And yeah, that may sound pretentious, but a lot of this stuff doesn't translate too easily into natural language.--Star trooper man 19:50, 6 October 2009 (UTC)
I think it is bounded but open. - π 10:20, 30 October 2009 (UTC)
Those two are incompatible, at least in my understanding of the words from geometry.--Star trooper man 12:11, 30 October 2009 (UTC)
It is actually a rather thorny question. Inflation theory states that the universe expanded rapidly, perhaps faster than the speed of light, so the universe we see is limited by the distance light has traveled. Hence it appears finite but boundless as the further light travels, the seemingly greater the universe appears to be expanding. Interesting things like this - sort of thing, dubbed "dark flow" suggests that there are further reaches beyond which our observation allows. AceMcWicked 11:15, 30 October 2009 (UTC)
Sometimes I wish I'd studied GR back when I was an undergrad...--Star trooper man 12:11, 30 October 2009 (UTC)

More article[edit]

The next section I think this article needs is maybe a discussion on uncountable numbers and a very brief discussion of the continuum hypothesis. - π 12:14, 30 October 2009 (UTC)

Good ideas, I'll have a crack at some stage soon...--Star trooper man 12:40, 30 October 2009 (UTC)
I know nothing about this (infinity maths), but aren't there several different infinites using the Hebrew(?) alphabet to differentiate? I am eating Toast& honeychat 12:46, 30 October 2009 (UTC)
An infinite number; have a glance at the article now.--Star trooper man 12:54, 30 October 2009 (UTC)
Hee! Thanks (I think), I understand it notTh unsure.gif, but I'll take your word for it! I am eating Toast& honeychat 13:00, 30 October 2009 (UTC)
Nice, however if you have confused Toast you might need to crank the language down a notch. We are not writing encyclopaedias here. Could you explain what one-to-one correspondence is or find a convenient sized lie to put in its place? What we need now is a woo/misunderstandings section. - π 13:08, 30 October 2009 (UTC)
Actually have found a place for an explanation. Please improve if you can. - π 13:13, 30 October 2009 (UTC)
I have to confess that I know little about maths and, having read what I assume to be a simplified description of infinity, I realise that I understand even less that I thought I did. From my uneducated point of view it might as well be written in Latin.--BobNot Jim 17:03, 30 October 2009 (UTC)
Which bits? Let us know so we can improve to make it more accessible.--Star trooper man 17:11, 30 October 2009 (UTC)
OK. I understand the introduction to here:
  • A set is countably infinite or denumerable if, for any n\in\mathbb{N} (i.e., any non-negative integer) there is a corresponding unique member of the set. An uncountably infinite set, like the real numbers \mathbb{R}, is even larger.
Presumably there is a difference between "infinite" and "countably infinite" though I don't know what that is.
Countably infinite means that you can put the elements into one-to-one correspondance with the natural numbers.--Star trooper man 17:30, 30 October 2009 (UTC)
I think that's what it should say, then. The math symbols in this particular case merely obscure a simple English statement IMO. ħumanUser talk:Human 17:58, 30 October 2009 (UTC)
Further, uncountably infinite means that you can't map the elements of the set to the natural numbers. Counting, loosely, means having or being able to construct an algorithm for mapping the elements of any particular set to the natural or counting numbers.--Star trooper man 17:36, 30 October 2009 (UTC)
Why is it important that the integer be non-negative - presumably the set of negative integers is also infinite.
It isn't important, it's convenient.--Star trooper man 17:30, 30 October 2009 (UTC)
 I don't understand what the end of the sentence "a corresponding unique member of the set" means.
That for every a in set A there is one b in set B to pair it to, and that no a has to share a b.--Star trooper man 17:30, 30 October 2009 (UTC)
I don't understand the term: "An uncountably infinite set" 
Above.--Star trooper man 17:30, 30 October 2009 (UTC)

and the term \mathbb{R} is unfamiliar to me.

The real numbers.--Star trooper man 17:30, 30 October 2009 (UTC)

Apart from that I understand the introduction quite well.--BobNot Jim 17:24, 30 October 2009 (UTC)

For the non mathematicians among us (which I suspect is the majority): why do we have to call them by strange symbols with suffixes like:(0) and +? Can't we call them Fred and Margaret or something? I am eating Toast& honeychat 20:26, 30 October 2009 (UTC)

Incompleteness[edit]

I like the article, but I don't think the statement "Unfortunately, owing to Godel's incompleteness theorem, this question is unanswerable." is really true. Incompleteness shows that statements like this exist. It's a separate result of Cohen (which does not require incompleteness) that the continuum hypothesis is actually one of these statements (in the sense that it's independent of ZFC). --Pyfgcr 17:19, 30 October 2009 (UTC)

I think you're right, and I'll have a look at that at some stage, unless a real mathematician comes along first...--Star trooper man 17:31, 30 October 2009 (UTC)

Delete (or at least reform)[edit]

This article needs to be deleted, or at least completely re-written from a different perspective. It is inconsistent with the statement of purpose at RationalWiki. Moreover, it is unfocused and simply wrong. It contains several mathematical errors (such as the claim that the Continuum Hypothesis follows from Godel's Incompleteness Theorem). Rationalwiki is not an encyclopedia. Wikipedia and other sites do a fairly good job of explaining mathematical and scientific ideas. See here for Wikipedia's Infinity article. It's not perfect, but it hits most of the main points. Note how many auxiliary articles are needed to explain all of the mathematical concepts surrounding "infinity." Is that really a project we want to take on?

What this article should be is a listing of all the fallacious appeals to infinity by crackpots and the misuse of the mathematical idea of infinity for ideological reasons. Something along these lines.-Antifly 20:28, 30 October 2009 (UTC)

Here, here! I supposeCry2.gif. But it's interesting & has taken me to things that I never knew existed. Since reading it I've been all over t'web looking at infinity. I am eating Toast& honeychat 20:42, 30 October 2009 (UTC)
I agree in the sense that we needn't go in depth with the concept of infinity, but more toward revealing misuse by those who don't understand. We should, however, at least leave some background information about the actual concept.
By the way, welcome back! Aboriginal Noise with 4 M's and a silent Q 20:47, 30 October 2009 (UTC)
RW might not be an encyclopedia, but pop-science and pop-maths articles to act as a background to the more on mission stuff is essential. Especially as Wikipedia has a very dry and sometimes very inacessible style. I say that there is no reason at all to delete this. If there are errors, correct them. Scarlet A.pngsshole 20:51, 30 October 2009 (UTC)
Knew there was a reason for keeping it. I am eating Toast& honeychat 20:54, 30 October 2009 (UTC)
True, true. Perhaps expanding the "Misapplications" section could add that much needed snark? Aboriginal Noise with 4 M's and a silent Q 20:55, 30 October 2009 (UTC)
Let's take that little section at the end and move it to the top (and pump it up a bit), since it's really the "RW Article", but leave all the technical stuff below it? ħumanUser talk:Human 20:56, 30 October 2009 (UTC)

I have made a start at re-writing this "article." Feel free to jump right in! I don't agree with Human's suggestion. The article is called "Infinity" for better or for worse, and should begin with an explanation of infinity. We could re-name it "Abuse of Infinity" an re-order the sections, or create a new article "Abuse of Infinity" that links here. Also, the misuse section is in dire need of improvement. Surely we can do better than the examples currently given...-Antifly 05:17, 31 October 2009 (UTC)

Also, I know very little about the use of infinity in physics, philosophy and theology. Perhaps our resident experts (or at least journeymen) can fill those sections out.-Antifly 05:19, 31 October 2009 (UTC)
I still think the "abuse" section should be prominent, before all the math. The lead should do a decent "math tag" free description, then the abuses, then the delightful geek stuff. ħumanUser talk:Human 06:07, 31 October 2009 (UTC)
OK, I just looked at it again, and this article is far worse than it was 12 hours ago. Then, it was actually pretty good, and possibly even understandable and interesting to the layman. Now it's just gibberish, concepts introduced out of order and poorly explained, etc. I am sorely tempted to revert back to a readable version. Oh, and I cut a bunch of empty headers, I would have pasted them here but I lost them. Empty headers are for sandboxes. Antifly, you have wrecked was was on the way to being a decent, readable article. It is now once again full of undefined, unclear, and unnecessary math symbols, and flows like a glassful of broken razor blades. I'll leave it alone for now, but I am sorely tempted to revert back to before this change, which is when it became undecipherable. Perhaps we should discuss phrasing and major changes here? Maybe? PS, I disagree with your arguments about the title versus the content. It's called "infinity", but it is still an RW article, not a definition of the term in encyclopedic language. ħumanUser talk:Human 06:16, 31 October 2009 (UTC)
I agree with Human. I've not worked through all the revisions and my comments refer to the earlier "mathematical" versions of the article. We can arbitrarily divide the world into two types of people. Those who understand mathematics and mathematical symbols and those who don't. In terms of our mission I suspect that those who understand maths are the least likely to be taken in by woo and pseudoscience. Those who don't understand mathematical symbols won't read anything that contains them. Consequently:
  • Those who understand maths don't need it.
  • Those who don't understand maths won't read it.
What we need then is a symbol-free version which talks about how infinity is misused in pseudoscience. If infinity is not used in woo and pseudoscience then we don't need the article in any form.--BobNot Jim 07:51, 31 October 2009 (UTC)[http://tap3x.net/EMBTI/j8gonsowski.html
I slightly agree, but maths is used in woo extensively, the question is of how notable woo has to be for us to mention it. This is pretty bad, but I'm not sure it counts as notable. Perhaps we need some notability guidlines and/or a blanket policy for dealing with maths articles? What do you guys think.--Star trooper man 07:58, 31 October 2009 (UTC)

Sorry for wrecking the article. I may have jumped the gun, so I reverted back to the old version. But to say that the article was better before really gets to the heart of why mathematics is abused by cranks. Sure the article was easier to read before. That's mostly because it was wrong. In order to make it correct you have to add math (you know undefined, unclear, unnecessary symbols). That was exactly my point in beginning this section. Look at the Wikipedia article, look at how many auxiliary pages are needed to explain all of the jargon. That's why we shouldn't be writing encyclopedia articles.- Antifly Now with 50% less retirement! 15:10, 31 October 2009 (UTC)

It's a good point tbh. Whilst I don't think the article before (or in it's current state) is terrible, it's not great either. And yes, to whomever removed it, the stuff I put in about Godel was misleading, so sorry. I think most of my other changes were positive though.--Star trooper man 16:07, 31 October 2009 (UTC)
I'm starting to wonder if we need two different articles.--BobNot Jim 18:16, 31 October 2009 (UTC)
I've scribbled some notes on a piece of paper I may try to use to fix this up a little without removing anything. Hopefully motivation will strike shortly... ħumanUser talk:Human 21:47, 31 October 2009 (UTC)

Problems With the Article[edit]

Here is a list of problems with this article:

I largely agree, I've added a few remarks.

1. The first sentence is completely inaccurate. Infinity is not a number and certainly isn't "bigger than that." There are many different infinite objects in different areas of mathematics. The only objects I can think of that could reasonably be called "infinity" are the elements +\infty,-\infty in the extended real line. But to suggest that infinity only arises in this context, or in the context of numbers in general is wrong. If we want to explain "infinity" in mathematics we need (at least) a discussion of: cardinals, ordinals, and convergence/divergence.

I agree.

2. No mention is made of the idea of infinity outside mathematics.

True, though I'm not so convinced that's a problem.

3. The fact that it is "still rejected by some people" needs a cite and further explanation (I don't really know who is rejecting what).

That is a problem indeed.

4. I'm not a physicist, and could be wrong here. Have we really shown that time and space are finite?

I'm not convinced that we have, though I'm no expert in GR.

5. Second paragraph, first two sentences: What?

Doesn't seem too bad.
Seeing as there is no such thing as infinity, starting the paragraph by purporting to explain what it is, is silly. Especially considering that no definition is ever given. Then we claim to define infinity as the opposite of finite, which it isn't. The opposite of finite is infinite.

6. Why does the article, from the 2nd paragraph onward, only focus on infinite cardinals?

Good point.

7. Second paragraph, last three sentences: This is blatantly wrong. The idea of an uncountable set has been conflated with the idea of a dense order which has, rather bizarrely, been conflated with the ability to list things.

Can you explain what's wrong with it? I thought that was how we defined an uncountable set - if an injection from a set to the natural numbers is possible, the set is countable. Otherwise, not.
What you just said is true. What the article says is not true. An uncountable set is a set that is not countable. A set is countable if there is an injection from the set to the natural numbers. (As you rightly point out).
What the article says is that an uncountable set is a set that is impossible to list, "there is no "next" real number, so they can't be listed." Unfortunately, the concept of "next" is outside set theory (which is where we are operating). In order to talk about "next" you need to be talking about a total order (or a partial order with a suitably modified definition of "next"). It is true that if we take the standard ordering of the real numbers, a real number x has no successor. But that is also true of the standard ordering of the rational numbers, a countable set (simply note that if x and y are rational, there is a rational number z between them). This property is called density and is a property of total orders, or topological spaces, but not of sets per se. Also, the inability to "list" a set is ambiguous. If by "list" you mean construct an injection to the natural numbers, then inability to list is equivalent to being uncountable, but only because you have replaced the word "countable" with "list" (it simply muddies the explanation). Once you explain what is meant by "list" we can resume the discussion, but you may want to take a closer look at transfinite ordinals first.

8. The "Now it gets tricky" section is pretty good for a six sentence introduction to convergence. But it is unclear how it is supposed to relate to the material before or after it, or to infinity in general.

It doesn't, scrap.

9. The "Counting" section could be presented better, but again is pretty good. It is again unclear why we have skipped from cardinality to convergence and back to cardinality.

Hmmm...

10. The "Now Counting gets tricky" section should be deleted. It consists of jargon, and a fairly poor illustration of why there are as many non-negative rational numbers as there are natural numbers.

Probably, or at least needs to be clearer.

11. "The paradox rests ...": There is no paradox. Infinite mathematical objects can have counter-intuitive properties.

I disagree that it isn't a paradox. A paradox is when a language is stretched outside of the domain to which it is usually applicable. Ideas of size for finite objects are clearly outside of their proper domain when stretched to infinite sets.
I suppose you are technically right. But I don't like using the word "paradox" while explaining math to a lay audience, as the alternate definition of paradox implies the existence of a logical contradiction. This leads people like Andy-pants to reject the Axiom of Choice because of the Banach-Tarski Paradox, without trying to understand what either actually means. Counter-intuitive is a better "word" for what we are talking about.

12. The "Cardinality, the reals, and the continuum hypothesis" is entirely jargon.

Yep, needs rewriting.

13. The final sentence in that section is still wrong. The idea of independence has been conflated with consistency. All that can be said is that the continuum hypothesis is independent of the ZFC axioms. That means that there is no proof of the continuum hypothesis assuming only the ZFC axioms. It says nothing about whether or not ZFC is consistent, nor about whether or not ZFC+(continuum hypothesis) is consistent.

Agree.--Star trooper man 05:52, 1 November 2009 (UTC)

I maintain that this article needs a complete rewrite, but I am happy to discuss fixes to these problems here.-- Antifly Now with 50% less retirement! 23:45, 31 October 2009 (UTC)

Apparently I didn't do my divinely inspired version yet (which may, of course, end up reverted without comment), but I have a germ of an idea of how to proceed here. Unless someone else beats me to it, of course ;) ħumanUser talk:Human 00:12, 1 November 2009 (UTC)

Fodder![edit]

This is probably the type of thing we are looking for. I will take a look when I have a chance. You probably need to go to a university (or even better be a student) to access the file.-- Antifly Now with 50% less retirement! 00:46, 2 November 2009 (UTC)

Oh you can't read it. Do you want me to send you a copy? - π 02:17, 2 November 2009 (UTC)
I can read it, thanks. I was just warning non-student types that they may not be able to access it...-- Antifly Now with 50% less retirement! 02:19, 2 November 2009 (UTC)
I agree. This crap and the "misuses" needs to be the first header, IMO. Then we do fun maths below that. ħumanUser talk:Human 02:02, 2 November 2009 (UTC)
You actual need the maths (particularly the cardinality stuff) to understand what is wrong with the article. - π 02:07, 2 November 2009 (UTC)
Yes, but it's ok for the explanation to be below the stupidity. Articles need not be read in a linear fashion? ħumanUser talk:Human 02:11, 2 November 2009 (UTC)
True, but it is a lot harder to read, as you can not go back to the spot you were reading after a detour. - π 02:14, 2 November 2009 (UTC)
How hard is it to scroll back up? Or implement inter-article links? Really? How hard is it to write "for more mathy explorations, see below" and "back to the snark", with links? ħumanUser talk:Human 02:33, 2 November 2009 (UTC)
You are assuming everyone stops reading and scrolls down at the same place. - π 02:56, 2 November 2009 (UTC)
No, I am positing that we can use "anchor" tags that send people back and forth. ħumanUser talk:Human 05:20, 2 November 2009 (UTC)

Density and non-discreteness[edit]

As mentioned above, countability is being conflated with "density". However, unlike what is pointed out above, density is the wrong word. Density applies to subsets; the appropriate term is, I believe, "non-discrete". — Unsigned, by: Imarcuson / talk / contribs Now signed by Imarcuson (talk) 06:39, 14 May 2010 (UTC)

Revisions[edit]

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. Scarlet A.pngsshole 18:03, 27 March 2011 (UTC)

If the universe is infinite[edit]

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. ħumanUser talk:Human 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. ħumanUser talk:Human 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[edit]

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)