Clarifying Infinity

[davidm]

You have two marbles. You have a foot-long ruler. You measure, with the ruler, the distance between the two marbles. You find they are one foot apart.

Some time passes, and you make a second measurement with the same ruler. You discover that the marbles are now two feet apart. You notice, also, that they are the same size as they were when you made the original measurement.

The distance between the marbles has doubled in size. But note that if everything else had also doubled in size, including you and your ruler, then in that case, you would still judge the marbles to be one foot apart.

But that is not what you see. So it is clear that you and your ruler have not doubled in size.

What are you to make of this?

You conclude, quite reasonably, that the distance between the marbles has doubled!

What’s the alternative? That you, your ruler, and everything else except the distance between the marbles, has shrunk one-half in size! Is that a reasonable thing to think?

Applied to the universe as a whole, it would mean that we, our measuring devices, and everything else, are shrinking in size except for the distances between galaxies, which are remaining static. If one were to entertain such an absurdity as a logical possibility, it is not a physical possibility. If the universe were static, Olbers’ paradox kicks in, not to mention universal gravitational collapse. We do not see either.

[someguy1]

Ok you know what, I'm perfectly correct.

From the Wiki page for the expansion of the universe, and I quote:

Technically neither space, nor objects in space, move. Instead it is the metric governing the size and geometry of spacetime itself that changes in scale.

That's exactly what I said. And it's not that deep a point. Distorting space is exactly the same thing as keeping space exactly the same, but changing your coordinate system or metric. It's just a point of view. When you say, "Oh but space is really stretching, like a rubber band," you're letting your intuition diverge from the science. Relativity is about coordinate systems. All the rest is metaphysics.

By the way on the point of the claimed clumpiness of gravity, that galaxies behave differently than planets, I'm still curious for an explanation.

[TheVat]

I'm still curious what my infinitely long iron bar does when it's heated. See my two posts that touched on that. Would this changing the metric concept also address the apparent expansion of an infinite bar?

[Positor]

I'm still curious what my infinitely long iron bar does when it's heated. See my two posts that touched on that. Would this changing the metric concept also address the apparent expansion of an infinite bar?

It would presumably become less dense. But that seems paradoxical, because it would remain the same length (i.e. infinite), with the same number of molecules, so its length divided by the number of molecules (and hence the distance between adjacent molecules) ought to remain the same. So how could its density change?

There is also the question of whether "infinitely long" has any real physical meaning.

[someguy1]

I'm still curious what my infinitely long iron bar does when it's heated. See my two posts that touched on that. Would this changing the metric concept also address the apparent expansion of an infinite bar?

I went back and found this one post. I'll try to address it but I'll try to be brief. I wrote a lot this afternoon already.

Our electron orbitals aren't moving away from our quarks and rendering our bodies into cyclotron trash.

Well it's a hell of a good question IMO to ask why they're not.

Of course in the case of subatomic particles, the answer is obvious. At tiny distances, electromagnetic and nuclear forces greatly predominate over gravity. It's obvious why quarks are immune to distortions in the gravitational field. The gravitational attraction between quarks is dwarfed by the nuclear forces by many many orders of magnitude.

It is NOT at all obvious -- or even, in my personal opinion, plausible -- that planets should behave differently than galaxies. I would love to understand this physics, if this is what's being claimed.

I don't really see any logical contradiction for cosmological scale expansion in an infinite universe, because infinities seem eternally accommodating.

You said this earlier with respect to the beachballs. You said you were not referring to mathematical infinities but then I know of no other. So I am still confused by your claim that "infinities seem eternally accommodating." I don't know what this means. It can't be just the Hilbert hotel. There are no "new guests," it's just a popularization of the fact that an infinite set can be bijected to one of its proper subsets. I am not following your point about infinities accomodating beachballs.

Only something with a limit cannot expand,

Mathematical counterexample. The sequence 1/2, 1/4, 1/8, 1/16, ... has a limit of zero. Suppose I "double the distance" between each consecutive pair of points by multiplying each term by 2. So now we have 1, 1/2, 1/4, ... You will see that we have term-by-term DOUBLED THE DISTANCE between every single point and its adjoining neighbors. Yet the sequence still has the same limit, namely zero.

So your statement is imprecise and unclear as to what it means. Something with a limit can expand, I just gave an example. And note something else. The first sequence 1/2, 1/4, ... is bounded within a universe of size 1/2. The double universe lives in an interval of length 1. In other words the size of our universe doubled. Finite universes double in length when you double their distance function.

so I need to understand what the "limit" would be for an infinite universe.

That's @davidm's territory, I have no idea what that would mean. But talking about limits is always trouble because limits have a very specific meaning in math and none at all in physics.

If you had an iron bar of infinite length, and you heated it up, what would happen to it?

Interesting question. Heat propagates at finite speed so if you heat one end (or the middle, doesn't really matter) as long as you apply heat, the temperature will increase gradually going outward. If you keep applying heat, the temp increase goes farther and farther out. But it never reaches infinity. Every point will eventually get hot; but the infinite outer part of the bar will always be cold. The border just keeps moving outward.

In other words: For any point x, it will eventually get hotter. But at time time t, only finitely much of the bar has gotten any hotter than it was before, and the infinite end (or ends) are the same temp as before.

This is the difference between unbounded and infinite. The heat increase is unbounded, but is always finite. It's just as the sequence of natural numbers 0, 1, 2, 3, 4, ... is unbounded, because eventually the sequence gets as large as you like. But every natural number is finite. There are no two natural numbers whose mutual difference is infinite.

That's how heat distribution must work, even on an actually infinite iron bar. As long as the propagation speed is finite.

Did you want to consider the case of infinite propagation speed? Then the entire bar will get hot right away.

Is any of this helpful? I'm not sure I understand the point about the iron bar.

[davidm]

Ok you know what, I'm perfectly correct.

From the Wiki page for the expansion of the universe, and I quote:

Technically neither space, nor objects in space, move. Instead it is the metric governing the size and geometry of spacetime itself that changes in scale.

Not only is the above in perfect accord with everything that I have said, the rest of that page also supports what I have said, including, please note, the following:

However, the model is valid only on large scales (roughly the scale of galaxy clusters and above), because gravitational attraction binds matter together strongly enough that metric expansion cannot be observed at this time, on a smaller scale. As such, the only galaxies receding from one another as a result of metric expansion are those separated by cosmologically relevant scales larger than the length scales associated with the gravitational collapse that are possible in the age of the universe given the matter density and average expansion rate.

[TheVat]

"Did you want to consider the case of infinite propagation speed? Then the entire bar will get hot right away.

Is any of this helpful? I'm not sure I understand the point about the iron bar." -Someguy


Well, sort of. My thought experiment was not notably reality based, but I was trying to get at an expansive effect all along the length of the bar, not at one location. If a uniform heat (yes, I know this requires infinite energy input) were applied, as Positor noted, it would become less dense and, erm, LONGER. But, of course, the bar is already infinitely long. Maybe this just reveals the failure point of preposterous thought experiments. In any case, I am very glad you have made clearer the distinction between actual stretching and changing the metric.

[TheVat]

I was briefly tempted to rename this thread by quoting Buzz Lightyear. You may recall his catchphrase was "To infinity...and beyond! "

[davidm]

From the wiki page:

A useful visualization is to approach the subject rather than objects in a fixed "space" moving apart into "emptiness", as space itself growing between objects without any acceleration of the objects themselves. The space between objects shrinks or grows as the various geodesics converge or diverge.

I am in complete agreement with this. With the whole article, in fact. In an infinite space, no object is moving apart into "emptiness." There is no emptiness, just an infinite space with an infinite number of objects, wherein over time distances increase between those objects.

[davidm]

Just to get clear, here are the credentials of the author of the "dot page" that SG disparaged.

And no, pre-emptively, spare me the charge that this is an illicit "appeal to authority." It's nothing of the kind.

[someguy1]

Just to get clear, here are the credentials of the author of the "dot page" that SG disparaged.

And no, pre-emptively, spare me the charge that this is an illicit "appeal to authority." It's nothing of the kind.

Why are you on about this? You haven't responded to Wiki quote I gave supporting my position; and you haven't explained why galaxies behave differently under gravity than planets do.

I am in complete agreement with this. With the whole article, in fact. In an infinite space, no object is moving apart into "emptiness." There is no emptiness, just an infinite space with an infinite number of objects, wherein over time distances increase between those objects.

You clearly didn't understand the Wiki page. Read the quote again. It's a change of coordinate systems or a change of metric, same thing. The "distance" changes only because you've changed the metric. You almost got it with everything but your final clause. You can THINK about it as space "stretching," but the math doesn't support that interpretation. The math just says the metric changes. You are making an interpretation, but it's not the only one possible. That's true even in the finite case, but in the infinite case your claim makes no sense. How can all the distances double yet the space remains the same size?

Not only is the above in perfect accord with everything that I have said ...

Wow. The quote I gave totally falsifies everything you've said. I've made my point and satisfied myself that I'm perfectly correct about the coordinate systems. This is the second time you were shown to be wrong and claimed to have had the opposite belief all along.

Re your Wiki quote: "However, the model is valid only on large scales (roughly the scale of galaxy clusters and above), because gravitational attraction binds matter together strongly enough that metric expansion cannot be observed at this time, on a smaller scale." ...

That explains why quarks aren't affected. It doesn't explain why planets aren't affected. You simply have not and can not explain this point.

[davidm]

How can all the distances double yet the space remains the same size?

Very easily, in infinite space. It's only in finite space that such cannot happen. In finite space, doubling in distance will run up against a limit.

Wow. We really are done here.

Yes, we are.

The quote I gave totally falsifies everything you've said.

The article supports everything I have said.

[someguy1]

Very easily, in infinite space. It's only in finite space that such cannot happen. In finite space, doubling in distance will run up against a limit.

That's shockingly incorrect. If I double all the distances in a one-inch interval it becomes a two-inch interval. You can't change the size of an infinite universe. You are simply making stuff up now.

Yes, we are.

Well you know what you can do with that Nathan's hot dog.

The article supports everything I have said.

Wow.

[someguy1]

Well, sort of. My thought experiment was not notably reality based, but I was trying to get at an expansive effect all along the length of the bar, not at one location. If a uniform heat (yes, I know this requires infinite energy input) were applied, as Positor noted, it would become less dense and, erm, LONGER. But, of course, the bar is already infinitely long. Maybe this just reveals the failure point of preposterous thought experiments. In any case, I am very glad you have made clearer the distinction between actual stretching and changing the metric.

Oh you are not asking about heat distribution, but rather about expansion. I get it. Well if you apply the map f(x) = 2x to the real line, the length of the line remains the same and each point has been relabeled. I don't know any other way to visualize this. You can't stretch an infinite line, it's already been stretched. You can THINK about the line as if it were being stretched, and I agree that this is a perfectly usual interpretation. But mathematically, the formula f(x) = 2x does not distinguish between stretching the space or scaling the coordinate system. As QM taught us, the equations are science and the interpretations are metaphysics. That would be my take.

[davidm]

Very easily, in infinite space. It's only in finite space that such cannot happen. In finite space, doubling in distance will run up against a limit.

That's shockingly incorrect. If I double all the distances in a one-inch interval it becomes a two-inch interval. You can't change the size of an infinite universe. You are simply making stuff up now..

Good lord.

Tell me, if I double infinity, what do I get? Or what does infinity times infinity equal?

The linked article supports everything I have said.

[someguy1]

Tell me, if I double infinity, what do I get? Or what does infinity times infinity equal?

If x is a cardinal, then x^2 = x. That's a theorem. If x is an ordinal, it's more complicated. Which do you intend to ask about? I strongly suspect that you haven't studied infinity enough to think clearly about it. That's evident in your posts.

https://en.wikipedia.org/wiki/Cardinal_ ... arithmetic

https://en.wikipedia.org/wiki/Ordinal_arithmetic

The linked article supports everything I have said.

You said the universe stretches. The article, and the math, say that the metric changes and that nothing in the space moves. It's clear as day that the article says the opposite of what you've been saying.

[davidm]

Interesting how you did not address this -- precisely what I have been saying:

However, the model is valid only on large scales (roughly the scale of galaxy clusters and above), because gravitational attraction binds matter together strongly enough that metric expansion cannot be observed at this time, on a smaller scale. As such, the only galaxies receding from one another as a result of metric expansion are those separated by cosmologically relevant scales larger than the length scales associated with the gravitational collapse that are possible in the age of the universe given the matter density and average expansion rate.

Now to go back to the author of the dot page, which you so smugly disparaged.

You can think that some rando on the internet like me is an idiot. Do you also think the author of that page, which agrees with everything I have said, is also an idiot?

[davidm]

You said the universe stretches.

I did not say any such thing. Perhaps you are getting this from a science chat forum to which I linked, where an interlocutor invoked an analogy of a piece of rubber stretching. It was an analogy. I, myself, said no such thing.

I'm really tired of your twaddle.

[someguy1]

I did not say any such thing. Perhaps you are getting this from a science chat forum to which I linked, where an interlocutor invoked an analogy of a piece of rubber stretching. It was an analogy. I, myself, said no such thing.

Oh so now I'm not supposed to read your links? When I criticized one of your links you've been vociferously defending it ever since, from listing the credentials of the author to accusing me of being against the WMAP project because I called out the obvious howlers in the article. Your track record's not good here. From now on when you post a link make sure to tell me which parts you don't even believe yourself.

I'm really tired of your twaddle.

How do you like that hot dog up your ass? I notice you didn't respond to my comments on cardinal and ordinal arithmetic. Embarrassed to have your ignorance revealed for all to see? This really the way you want to go? I can go as low as you can, friend. You've been snotty for pages and I've had enough. Not only just to me but to others as well. I wonder if it's crossed your mind that you have no idea what you're talking about. You don't even understand the pages you're linking because none of them say what you claim they do.

[Positor]

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.

The galaxies (and local groups of galaxies) do not expand. That, I understand, is an observed fact. So the 'dots' in this case remain the same size; the 'balloon' is an incorrect analogy.

The question is: why does gravity counter the cosmological expansion, up to a particular distance but no farther? Why that particular distance? And how does gravity 'know' how to balance the cosmological expansion exactly? As you ask, how does the matter within each galaxy "know how to stay right where it is with respect to its neighbors"? But it clearly does. No doubt there is some mathematical formula linking the effects of cosmological expansion and gravity.

[TheVat]

....It's a change of coordinate systems or a change of metric, same thing. The "distance" changes only because you've changed the metric. You almost got it with everything but your final clause. You can THINK about it as space "stretching," but the math doesn't support that interpretation. The math just says the metric changes. You are making an interpretation, but it's not the only one possible. That's true even in the finite case, but in the infinite case your claim makes no sense. How can all the distances double yet the space remains the same size?

- SG

It seems reasonable to me that we can ask if this metric stretching effect is more pronounced in the void between galaxies where the gravitational forces are weaker than they would be at more "local" scales (inside of galaxies). In voids, the energy density of empty space, or "vacuum energy" would be the predominant player.

IF this latter possibility is logically inconsistent with an infinite space, then that could lend support to a "bubbleverse" kind of inflationary theory where Big Bangs occur constantly and inflation happens in a 4D bubble, i.e. in each universe, but not in the multiverse as a whole. In an infinite ocean, our cosmos would just be one expanding bubble of sea foam. In that theoretic framework, ALL inflation would be "local" (to that bubble) and not create a logical contradiction where inflation is happening everywhere.

[TheVat]

No more red-shifting Nathan's hot dogs, please.

[davidm]

[. I wonder if it's crossed your mind that you have no idea what you're talking about. You don't even understand the pages you're linking because none of them say what you claim they do.

Yes, they all say exactly what I say.

It is an empirical fact that distances are growing between galaxies. It is a fact that NASA WMAP has provided strong evidence for a spatially infinite universe. It is possible for an infinite space to contain infinite objects and for the distances between those objects to grow over time, even an infinitely long time. Infinity times infinity is infinity.

Bye-bye.

[davidm]

Just to reiterate for any late comers:

How can the Universe be infinite if it was all concentrated into a point at the Big Bang?

The size of the box in each view is 78 billion light years. The green circle on the the right is the part of the Universe that we can currently see. In the view on the left, this same part of the Universe is shown by the green circle, but now the green circle is a tiny fraction of the 78 billion light year box, and the box is an infinitesimal fraction of the whole Universe. If we go to smaller and smaller times since the Big Bang, the green circle shrinks to a point, but the 78 billion light year box is always full, and it is always an infinitesimal fraction of the infinite Universe.

Very clear. Bold by me.

[TheVat]

Just to reiterate: this is a math thread. Cosmological theory should be approached from the perspective of what math concepts of infinity (Cantor, Dedekind et al) are applicable.

https://www.britannica.com/science/infinity-mathematics