Clarifying Infinity

[ronjanec]

Crimony, I went back to page one, and see that Someguy and Lomax* were explaining and clarifying beautifully back in April 2017. Newbies, please read the start of a thread before posting. Grrr.


*miss you, dude

I miss him too Biv. Though he always gave me a real run for the money with many of my posts here, he was always a really good moderator, and was always fair to all concerned.

[someguy1]

Once again, let us be clear: "doubling the size of the universe" means: the distance between objects doubles. It does not mean that space is an object that doubles in size.

Ok. Now I agree with that. Earlier I had the impression you said the opposite, that the size of the universe got larger too. But perhaps that was the page you linked me to, so if that was not your intent at least it's all clear now.


Anyway ... I think this is a very interesting and enlightening example. Let's kick it around.

Let's work on the real line again. If you prefer 2 or 3 dimensions that's fine, it's the same argument. We have a Euclidean space and its integer lattice. On the line, our galaxies are at the integer points ..., -2, -1, 0, 1, 2, 3, ... extending in both directions.

Now your visualization is that we "stretch the line" so that the integer points are now labelled ..., -4, -2, 0, 2, 4, ... I hope you can see that while we can think of this as stretching the line; we can ALSO think of simply walking down the line and repainting the address of each point. The point at 1 becomes the point at 2. The point at 2 becomes the point at 4. And so forth.

In other words a real number is the address of a point on the real line. It is not the point itself, which has no name. The real numbers are a particular coordinate system on the real line. If we apply the transformation f(x) = 2x to each point on the real line, the resulting output set is still the real line. We can THINK of it as "stretching the line." But it's EQUALLY ACCURATE to say that we are merely repainting the addresses. It's like the city council declaring that all the houses on your side of the street are getting a new address. You have to repaint the curb in front of your house, but nothing else in the universe changes. Only the label of a point changes.

How do you tell the difference between the before and after universe? Is the new line the "same" as the old one with just the coordinate system changed? Or has it somehow been "stretched" so that distances are now longer?

Well, since Einstein showed that there is no preferred frame of reference in the universe, the question is meaningless. It's axiomatic in physics that changing the coordinate system does not change the world.

So when we apply the transformation f(x) = 2x to each point of our line-universe, it makes more sense to realize that all we've done is introduce a new coordinate system. Nothing else has changed. We've just renamed each point, but everything's exactly where it was before. In the absence of a fixed background frame of reference, there is simply no way to distinguish the two cases.

In short, you cannot claim that applying a linear scaling factor to your coordinate system makes a material change in the real world. It doesn't, anymore than repainting the numeral on the curb in front of your house changes the location or nature of your house.

I will grant you that the distance has doubled between neighboring pairs of our distinguished points, our galaxies that were formerly at the integer positions and that are now at the even integers. But I'm not sure what that means or why it matters. I know it's important to your argument, but I don't understand why.

On the other hand our original one-unit measuring rod has doubled in size too, so the difference between neighboring galaxies is still one measuring rod. If everything in the universe stretches, your measuring rods stretch too. You can't tell the difference before and after. The before-universe and the after-universe look identical.

I'll leave it at that for now. I'd like to know your thoughts.

[davidm]

You’re still missing the most important point that I am trying to make here. With 100% no doubt metaphysical certainty, it is impossible for an infinite sized universe to increase in size, because if it could ever increase in size it was never infinite to begin with.

I am asking: is it, or is it not, possible for the distances between objects to increase with time in an infinite universe? If not, why not?

[ronjanec]

Once again, let us be clear: "doubling the size of the universe" means: the distance between objects doubles. It does not mean that space is an object that doubles in size.

This is what you said before: “An infinite universe”: Now you are saying that this does not mean that space is an object that doubles in size along with the infinite universe doubling in size? In an infinite sized universe, space would also have to be infinite in size, and would also have to double in size along a with an infinite universe, and that is again impossible.

[davidm]

I'm sorry, the above makes no sense to me. Again, I am asking a simple question.

Is it possible, in a spatially infinite universe, for the distance between any two objects to double in size from t1 to t2? If not, why not?

[ronjanec]

You’re still missing the most important point that I am trying to make here. With 100% no doubt metaphysical certainty, it is impossible for an infinite sized universe to increase in size, because if it could ever increase in size it was never infinite to begin with.

I am asking: is it, or is it not, possible for the distances between objects to increase with time in an infinite universe? If not, why not?

You are bouncing around all over the place David and keep changing the wording, and the premise of the earlier discussion;

Of course it is possible for the distances between objects to increase with time in an infinite universe, but you were earlier calling this an infinite number of implied galaxies doubling in the distance in size between them, all existing in an implied infinite universe, therefore doubling the size of the again implied infinite universe, and that is a completely different story(and again impossible)

[someguy1]

I'm sorry, the above makes no sense to me. Again, I am asking a simple question.

Is it possible, in a spatially infinite universe, for the distance between any two objects to double in size from t1 to t2? If not, why not?

Are you addressing me? Can't tell. If you didn't understand my post perhaps you could ask specific questions. It seems perfectly clear to me. There's no preferred frame of reference. A change of coordinates by a scalar factor (2 in this case) does not affect the underlying universe in the least. An observer would not see any difference between the before and after universes.

But can you just tell me what is your point? Suppose for sake of argument that I said ok, God has a fixed background coordinate system and relative to that, the universe has been stretched. What is your conclusion? I'm missing that part of your argument.

[davidm]

You are bouncing around all over the place David ...

No, I am not. I am sticking right to the point.

Of course it is possible for the distances between objects to increase with time in an infinite universe...

Thanks.

...but you were earlier calling this an infinite number of implied galaxies doubling in the distance in size between them, all existing in an implied infinite universe, therefore doubling the size of the again implied infinite universe, and that is a completely different story(and again impossible)

No it is not different. It is exactly the same.

If it is possible for the distance between two or more objects to double in size in an infinite universe, it necessarily follows that it is possible for the distance between an infinite number of objects to double in size in an infinite universe.

[davidm]

I'm sorry, the above makes no sense to me. Again, I am asking a simple question.

Is it possible, in a spatially infinite universe, for the distance between any two objects to double in size from t1 to t2? If not, why not?

Are you addressing me? .

I was addressing ronjanec. Should have quoted, sorry. Will get to your latest post tomorrow, though it will probably be quite late, as once again I am going to Coney Island. :-)

[someguy1]

I was addressing ronjanec. Should have quoted, sorry. Will get to your latest post tomorrow, though it will probably be quite late, as once again I am going to Coney Island. :-)

Thanks for clarifying. We have something in common. I love Coney Island. I'm on the west coast now so have a Nathan's hot dog for me.

[davidm]

I was addressing ronjanec. Should have quoted, sorry. Will get to your latest post tomorrow, though it will probably be quite late, as once again I am going to Coney Island. :-)

Thanks for clarifying. We have something in common. I love Coney Island. I'm on the west coast now so have a Nathan's hot dog for me.

I will! I have lived on the West Coast too and love the southern Cal beaches.

[TheVat]

SG: Just to clarify again, I wasn't saying the cardinality of all beachballs was the same as the real numbers, i. e. the cardinality of the continuum.

BIV, You said there's a beachball in the real numbers. There is not. Here's the quote: "Brent, an infinite space is a continuum and has abundant room for infinite sets of anything - atoms, beach balls, short bald men named Balthazar, you name it." That's false. There are no beachballs in the real numbers. Infinite sets don't necessarily have "room" in them. I hate to be picking on this but I really don't know what you mean.

But the cardinality of beachballs and bald guys is the same in an infinite multiverse, right?

No, why should it be? Maybe there are 5 bald guys and 7 beachballs. The notion that there must be infinitely many of everything in an infinite universe is false. Also, the infinite universe idea is statistical. There could be all sorts of measure zero anomalies. It's simply not true that "everything must happen infinitely many times." In the infinite sequence 010101010101... there is never a 2. In the infinite sequence 01111111... there are indeed infinitely many 1's but 0 never recurs. In an infinite universe where matter can only take a bounded number of states in a given region of spacetime, at best you can say is that SOMETHING recurs infinitely often. But not everything.

I think you omitted the part of my quote where I said I wasn't using continuum in the math sense, i. e. not real numbers. And I said potentially, not necessarily, in regards to infinite beachballs or bald men. Of course infinite beachballs is not a logical necessity, and I don't know how I gave such an impression. Forget real numbers. If there is infinite space, then there can be infinite number of finite objects of a certain class in that space. Countable thingies. Natural numbers. Potentially. That's all I was saying. Brent was saying that is not possible. I guess making physical observational comments is just breeding confusion in a math thread.

[davidm]

I will grant you that the distance has doubled between neighboring pairs of our distinguished points, our galaxies that were formerly at the integer positions and that are now at the even integers. But I'm not sure what that means or why it matters. I know it's important to your argument, but I don't understand why.

On the other hand our original one-unit measuring rod has doubled in size too, so the difference between neighboring galaxies is still one measuring rod. If everything in the universe stretches, your measuring rods stretch too. You can't tell the difference before and after. The before-universe and the after-universe look identical.

I'll leave it at that for now. I'd like to know your thoughts.

This is a completely different argument, and not at all what I’m talking about.

You are asking if everything in the universe instantly doubled in size, including ourselves and our measuring rods, would we be able to detect this change? It would seem not. This is an old philosophical hobby horse.

But the distances between galaxies are increasing all the time. Our measuring rods remain the same size. This is an objective fact of nature, and is called the expansion of the universe.

The only point at issue is whether distances between objects can increase in an infinite universe, containing infinite objects. Of course such distances can increase, and they can continue do so for an infinite time! Why in heaven’s name would they not be able to do this?

As I have noted, WMAP provides strong evidence that the universe is spatially infinite. I reiterate: Brent’s claim that the universe cannot be infinite, with an infinite number of objects, is refuted.

[TheVat]

SG,

I see the problem here...

In other words a real number is the address of a point on the real line. It is not the point itself, which has no name. The real numbers are a particular coordinate system on the real line. If we apply the transformation f(x) = 2x to each point on the real line, the resulting output set is still the real line. We can THINK of it as "stretching the line." But it's EQUALLY ACCURATE to say that we are merely repainting the addresses. It's like the city council declaring that all the houses on your side of the street are getting a new address. You have to repaint the curb in front of your house, but nothing else in the universe changes. Only the label of a point changes.

In a physical universe, you aren't just repainting addresses. The reason your labeling argument doesn't apply is because of the physical aspects of light - red shift, parallax shift along different lines of sight, absolute v. apparent magnitude of certain variable stars, etc. Space is real stuff that really stretches. This is where the math of continua may not quite work.

I see now how this chat went off the rails. Sorry I didn't earlier.

[davidm]

In case this is not clear, please note: the cosmological expansion is just that: cosmological. It is not happening everywhere. The distances between the atoms of our body are not increasing. The distance between the sun and Alpha Centauri is not increasing, etc. The distances between galaxies are increasing -- hence, "cosmological" expansion. Indeed, if the distances between every single thing, including our constituent atoms, were growing, and were doing so at the same rate, then we would not detect any expansion at all. We would judge the universe to be static.

[TheVat]

Yes, though for quantum mechanical reasons, a distance increase in electron orbitals would cause a breakdown of condensed matter that would destroy all biology. It would also require a major field strength drop in the Higgs field which would cause all baryonic matter to fly apart.

But your point is well taken. If scale changed in some absolute and homogeneous fashion, then it would be meaningless to look for any different metric in the physical world. In fact, I think "change in scale" would just be invalid as a concept, maybe. Come to think of it, the only way to a change in scale is to change the Planck length, which would mean moving to a different M-theory universe with different physical constants.

[ronjanec]

davidm,

When you are theoretically talking about an infinite universe supposedly doubling in size and extent for any reason, you must also be talking about a theoretically previous infinite quantity of space existing in this same universe doubling in size and extent(this is in spite of what you said earlier about the space in the infinite universe not also doubling in size), and that is again impossible, because it was said to have previously existed in an infinite state.

If a theoretical infinite universe could actually double in size for any reason whatsoever, then there was a limit to it’s size previously or before, and it was then never infinite to begin with.

[scientificphilosophe]

I have to agree with someguy, ronjanec and others that when something is infinite it is boundless. Doubling it is a false concept.

However the original point of the discussion was that the infinite can be directional - depending on your definition.
This point seems to have been lost - as has the need to be clear about set definitions and what they encompass.

To make the point again, the set of positive integers running from 6 onwards will still be infinite in the increasing direction, but it will be be finite and limited on the decreasing direction. It does therefore seem that either 'direction' or partial restrictions can still apply on an infinite set.

There is also a continuing need to differentiate between a potentially infinite set and the degree to which it is populated. That becomes even more important when we think that something entirely different to the original set is deemed to occupy it... ie if the original set is seen as a container made of entirely different material to any perceived contents.

There have been many posts which started discussing the possible 'infinity of space', (rather than the expanding physical universe within it) and extrapolating that infinity to any potential contents of an infinite space. The contents require a separate definition - they are not tied together.

Put more simply, an infinite space could have a finite physical universe within it... but equally the physical universe might be infinite too. We don't know, other than to say that when entering these debates both would need a separate 'set' definition.

[someguy1]

I think you omitted the part of my quote where I said I wasn't using continuum in the math sense, i. e. not real numbers.

Oh that makes perfect sense then. Beachballs! I did not realize you weren't talking about the mathematical continuum. Ah ... what definition of continuum are you using?

[quote="[url=http://www.sciencechatforum.com/viewtopic.php?p=339982#p339982]
Forget real numbers. [/quote]

Ok. The reals are the only continuum I know. Well I know a little about the constructive and intutitionistic continuum, and the hyperreals and surreal numbers at the other end of the scale. But all models of the continuum I know are mathematical. If there's a continuum containing a beach ball I'd be interested to know about that and what it means.

[someguy1]

In a physical universe, you aren't just repainting addresses. The reason your labeling argument doesn't apply is because of the physical aspects of light - red shift, parallax shift along different lines of sight, absolute v. apparent magnitude of certain variable stars, etc. Space is real stuff that really stretches. This is where the math of continua may not quite work.

I'm prepared to debate that point but I'll have to get to it tomorrow or day after. If space stretches do the molecules of my body move farther away from each other? That would not support the same biological processes.

Remember the point is that @davidm is talking about stretching an infinite universe. In that case I suspect people's physical intuition is breaking down and my real number model holds perfectly well.

As far as relabeling, isn't relativity all about seeing what's invariant under various changes of coordinates? A linear scaling factor in an infininite space can't possibly make any difference to anything. That's my sense of it, but I don't know the physics.

@davidm I'll try to get to your post later.

[davidm]

davidm,

When you are theoretically talking about an infinite universe supposedly doubling in size and extent for any reason, you must also be talking about a theoretically previous infinite quantity of space existing in this same universe doubling in size and extent(this is in spite of what you said earlier about the space in the infinite universe not also doubling in size), and that is again impossible, because it was said to have previously existed in an infinite state.

If a theoretical infinite universe could actually double in size for any reason whatsoever, then there was a limit to it’s size previously or before, and it was then never infinite to begin with.

I see nothing to respond to here, because this post makes no sense to me.

[ronjanec]

davidm,

When you are theoretically talking about an infinite universe supposedly doubling in size and extent for any reason, you must also be talking about a theoretically previous infinite quantity of space existing in this same universe doubling in size and extent(this is in spite of what you said earlier about the space in the infinite universe not also doubling in size), and that is again impossible, because it was said to have previously existed in an infinite state.

If a theoretical infinite universe could actually double in size for any reason whatsoever, then there was a limit to it’s size previously or before, and it was then never infinite to begin with.

I see nothing to respond to here, because this post makes no sense to me.

Well, I feel that you and I have come to a complete impasse in regards to this particular subject davidm, and we both need to move on from here: hopefully to a different topic that we can both agree on.

[TheVat]

I wonder if the core confusion here is regarding what precisely we mean by expansion of space. David is referencing a cosmological expansion, not a local one. Our electron orbitals aren't moving away from our quarks and rendering our bodies into cyclotron trash. I don't really see any logical contradiction for cosmological scale expansion in an infinite universe, because infinities seem eternally accommodating. Only something with a limit cannot expand, so I need to understand what the "limit" would be for an infinite universe. If you had an iron bar of infinite length, and you heated it up, what would happen to it?

I fully acknowledge the migraine-inducing aspects of this discussion, and hope Someguy, or some guy (heh), can help us map the relevant math concepts onto physical thought experiments.

[TheVat]

To be sure, the cardinality of the iron bar might that of natural numbers, if one holds that the Planck length is the discrete unit of space in a QT framework. We can assign ordinal numbers to each Planck length of the bar....first Planck, second Planck, third Planck, ad infinitum. The iron bar could not then be assigned points based on real numbers - you can't have an infinity of Planck lengths between O and 1. That means that all the numbers on the bar are a countable set. And what, must include zero, since the bar, wherever you stand, extends infinitely in both directions. You may as well call where you stand the zero point, with negative integers going one way and positive the other way. An infinite number of Planck lengths. And heating the bar will cause its atoms to get slightly farther apart in their metallic lattices, so the expansion will be adding Plancks - how can it not? Hmm.

[someguy1]

You are asking if everything in the universe instantly doubled in size, including ourselves and our measuring rods, would we be able to detect this change? It would seem not. This is an old philosophical hobby horse.

Ok. Then we agree on this technical point.

But the distances between galaxies are increasing all the time. Our measuring rods remain the same size. This is an objective fact of nature, and is called the expansion of the universe.

Right. Now my understanding of the expanding universe is that it's like concentric circles. We have a small circle that's the observable universe, and a larger outer circle that represents the whole universe. That's a simple fact. The claim that the outer universe is "infinite" is not supported by any science. I've looked at your links and I'm not convinced. One link you posted was of extremely poor quality.

The only point at issue is whether distances between objects can increase in an infinite universe, containing infinite objects. Of course such distances can increase, and they can continue do so for an infinite time! Why in heaven’s name would they not be able to do this?

If all distances double but the universe remains the same size, the the doubling amounts to nothing more than a change of coordinates.

I will grant you the point that if all distances doubled in size but my measuring rod remained the same, then I'd travel 3000 miles east of California, arrive in Omaha, and then I'd have to claim that Omaha is merely a relabeling of New York City. Of course I could never make that mistake, having been to Omaha and New York. But that's the measuring rod problem again so nothing's really gained. I also wonder what it means for all distances to double. Are my neurons twice as far apart? Can biological processes still work? I don't think so. This doubling you speak of is very murky and I would like you to explain the rules more clearly. You already concede my point if the measuring rods double along with everything else. So when you say that distances in an infinite universe double in size, how is that materially different than a mere change of coordinates?

As I have noted, WMAP provides strong evidence that the universe is spatially infinite.

I read your article and did not see that interpretation. I was not convinced.

I reiterate: Brent’s claim that the universe cannot be infinite, with an infinite number of objects, is refuted.

I am not addressing other posters here. I also don't believe that the universe can not be infinite. I said earlier that (a) I don't know if it's infinite or not; and (b) I find the evidence and arguments on both sides of the issue inconclusive and weak.

[someguy1]

I wonder if the core confusion here is regarding what precisely we mean by expansion of space.

I have most definitely come to that conclusion. @davidm agreed that if all distances double and the measuring rods double as well, we could not tell the difference.

And I've made the point that if my neurons and cells suddenly got twice as far apart, my biological processes wouldn't function properly. So this expansion has to be explained more clearly.

And the example of the simple high school function f(x) = 2x applied to the real line shows that the real line is exactly the same before and after, and all that's happened is that the points have been relabelled. I truly don't understand how it could be any different in an infinite universe. If all distances double but the universe is exactly the same size afterward (as @davidm agrees) then what does it mean for distances to have doubled?

I do agree that Omaha is not New York City so I have that doubt about my own argument. I'm not being dogmatic, I'm just confused by what is being argued.

I refer now to @davidm's awful link (he got upset at me when I mocked this link but it's impossible not to) here: http://www.astro.ucla.edu/~wright/infpoint.html

It says: Note that the black dots represent galaxies, and the galaxies do not expand even though the separation between galaxies grows with time.

REALLY? How does that work? How does the distance between two galaxies know to double, yet the matter within each galaxy knows to stay right where it is with respect to its neighbors?

We're often told to imagine dots on the surface of a balloon. But as the balloon expands, so do the dots.

[davidm]

I wonder if the core confusion here is regarding what precisely we mean by expansion of space.

I have most definitely come to that conclusion. @davidm agreed that if all distances double and the measuring rods double as well, we could not tell the difference.

Except that is not what is happening in real life, as i've explained, and will not explain again.

Since you seem to agree that the universe can be infinite, with an infinite number of objects, you now agree with me that Brent made a wrong argument when he claimed that this was impossible.

As to the WMAP link you scorned, do note that it was a link to NASA, that WMAP was a NASA project, and is considered one of the most successful NASA projects ever. That you, personally, don't find it compelling does not carry much weight.

The graphics page with the dots in expanding space is absolutely right. I'm sorry you can't grok it.

[someguy1]

Except that is not what is happening in real life, as i've explained, and will not explain again.

LOL I'm barely done editing. In general if you could wait a few minutes before snapping it would be helpful. I typically write a post then spend a little while tweaking it.

Also, when you say, ".. and will not explain again," is that to be interpreted as snipitude? I'd prefer civility and am making efforts in that direction. Would appreciate same. How can you be angry at someone you'd go to Nathan's with?

If you explained something six times and I don't get it, there are two possibilities: Either I'm an idiot, or you did not explain your idea as clearly as you think you did. I make no claims either way, but perhaps you might consider the second possibility as well as the first.

[davidm]

We're often told to imagine dots on the surface of a balloon. But as the balloon expands, so do the dots.

No, they don't. That's the whole point. The observed expansion is cosmological and not local.

[someguy1]

No, they don't. That's the whole point. The observed expansion is cosmological and not local.

Ah. Is it statistical? In other words something that's in the limit of some equation, but that may have many exceptions along the way?

I wonder if you can clarify this for me just on a technical basis. Can you tell me exactly what is being claimed? I'm perfectly willing to say I have a set with some metric and now you're going to introduce a new metric that models planets staying near each other but galaxies moving farther away. You could write down some function. So I can perfectly well live with this. Just explain to me how the galaxies know to double their distance but the planets don't.

I always thought I understood this, but apparently I don't. I thought there was a big bang and we live in the aftermath of the explosion. Or more properly, we are the explosion. For some reason the balloon used to make sense to me, but suddenly it doesn't. Are you saying that MACRO things get farther from each other, and MICRO things don't?

Please fill in the technical understanding I'm clearly missing here.

ps -- Didn't Newton prove that you can model gravity as if each body was a point mass? It was a great mathematical achievement in the service of a beautiful simplification of the physics.

Doesn't this challenge your claim? Galaxies and planets are point masses as far as gravity is concerned. They all behave the same. Gravity is universal. Newton says you are wrong. There can not be different gravitational behavior for galaxies than there is for planets.

I am not buying this. Please tell me what I'm missing here. If you are saying that relativistic or quantum effects make gravitation clumpy, this is something I have never heard of before.