Infinite Universe? Heaven?

[JerryZ]

Can you tell us why you think the bolded statement is true?

Well, 60 years ago, the prevalent idea about the Universe was that:
A) it was infinite in size and
B) it had existed forever.

Astronomy shattered both notions by measuring the red shift of the light arriving from all the observable galaxies around us. It pointed out that, they, ALL the galaxies we could observe were running away from us.

Meanwhile science and technology have improved and now we know that the Universe was formed 13.8 Billion Years ago (the Age of the Universe) and that it began with a release of energy from a singularity which promptly started expanding from the initial locus in all directions.

So if we can go back in time we see that the size of the Universe was NOT infinite.
In fact at t = zero the size of the Universe was also zero.
A singularity has no space and neither time.

The four dimensions were created at the instant the singularity decided or was pushed to change it’s state.

Now, currently the latest measurements suggest the Universe is a “flat” Universe that will go on expanding to infinite. But flat is really a mathematical model that tries to explain the behaviour of the Universe or how light travels through it. Are 2 parallel beams of light going to remain parallel or will they diverge away or toward each other AND can a beam of light travel forever away from the source OR will after a time = x return to the source.
It certainly does NOT mean that the Universe is like a sheet of paper with only 2 dimensions.

Hence current measurements favour that the light will move parallel forever. A “flat Universe”

A singularity by the way is one of the most difficult things to grasp by normal people. Because within the singularity “NOTHING” can exist, no time, no space, no matter.
And many people think of outer space or the vacuum of space as equivalent to the “no space” of the singularity.

The reality is that the “Interplanetary space” or “outer space” or vacuum of space" is not empty. Yes the density of the matter contained in it is quite low BUT it is not zero.

There is a LOT of matter in the Universe all around us.

…

 

[hecd2]

Well, 60 years ago, the prevalent idea about the Universe was that:
A) it was infinite in size and
B) it had existed forever.

Astronomy shattered both notions by measuring the red shift of the light arriving from all the observable galaxies around us. It pointed out that, they, ALL the galaxies we could observe were running away from us.

Meanwhile science and technology have improved and now we know that the Universe was formed 13.8 Billion Years ago (the Age of the Universe) and that it began with a release of energy from a singularity which promptly started expanding from the initial locus in all directions.

So if we can go back in time we see that the size of the Universe was NOT infinite.
In fact at t = zero the size of the Universe was also zero.
A singularity has no space and neither time.

The four dimensions were created at the instant the singularity decided or was pushed to change it’s state.

Now, currently the latest measurements suggest the Universe is a “flat” Universe that will go on expanding to infinite. But flat is really a mathematical model that tries to explain the behaviour of the Universe or how light travels through it. Are 2 parallel beams of light going to remain parallel or will they diverge away or toward each other AND can a beam of light travel forever away from the source OR will after a time = x return to the source.
It certainly does NOT mean that the Universe is like a sheet of paper with only 2 dimensions.

Hence current measurements favour that the light will move parallel forever. A “flat Universe”

A singularity by the way is one of the most difficult things to grasp by normal people. Because within the singularity “NOTHING” can exist, no time, no space, no matter.
And many people think of outer space or the vacuum of space as equivalent to the “no space” of the singularity.

The reality is that the “Interplanetary space” or “outer space” or vacuum of space" is not empty. Yes the density of the matter contained in it is quite low BUT it is not zero.

There is a LOT of matter in the Universe all around us.

…

This is (mostly) an unexceptional pop account of a few elements of the standard model of cosmology. Nothing here explains however why you believe your assertion “If it is expanding then A boundary exists.” is true.

 

[hecd2]

And although most of what you wrote is correct, this statement shows a lack of understanding of the physics and the properties of metric spaces:

JerryZ:

it began with a release of energy from a singularity which promptly started expanding from the initial locus in all directions.

Not correct:the expansion doesn’t proceed from an initial point; in fact, the initial locus is everywhere in space: cosmic expansion is an increase in the scale factor everywhere. The Big Bang happened everywhere.

 

[hecd2]

I liked yppop’s post.

You might have liked it, but that doesn’t change the fact that most of it was factually wrong.

What philosophical argument are you referring to that proves there is no boundary to our universe?

If the universe is all that physically exists, then it makes no sense to talk about “outside” it because the concept of a space outside the universe is meaningless. But a boundary implies a universe embedded in a bigger space.

 

[You]

You might have liked it, but that doesn’t change the fact that most of it was factually wrong.
If the universe is all that physically exists, then it makes no sense to talk about “outside” it because the concept of a space outside the universe is meaningless. But a boundary implies a universe embedded in a bigger space.

it makes more sense to talk about the universe existing within something than it existing in nothing - which would be sort of a contradiction.

 

[JuanFlorencio]

Well if you define a set so the contained elements are exclusive (in other words the definition excludes all other elements) then one can’t add new elements that satisfy the exclusive defintion, by definition. If I define an infinite set as follows: the set of all integers that contain the decimal digit 3: {+/-3,+/-13,+/-23,+/-30,+/-31,+/-32…} (which set, by the way, has the same cardinality as the natural numbers), then you can’t add an element that satisfies the definition of an integer which contains the decimal digit 3 because the set already contains all the integers that contain the decimal digit 3. That is trivially true for all exlusively defined sets.

I would like to see how you define a set which is not “exclusive”.

But that is not relevant to my argument which is about infinite sets which are not exclusively defined. For example, one can add the set of numbers which are the natural numbers plus 1/2 to the set you have defined to give a new set {1, 1 1/2, 2, 2 1/2}. For any countable infinite set which is not defined to exclude all elements that it does not contain, one can add any number of additional elements. As I have said three times already, this is simply Hilbert’s Hotel, which is a well accepted property of transfinite numbers. You can add another countably infinite set to any countably infinite set and get a countably infinite set with additional elements. In fact you can do that as many times as you like. Or you can just add one element.

aleph0 = aleph0+n
aleph0 = aleph0*n
aleph0 = aleph0^n
where n is any finite numer.

You started with the set of real numbers; then you wanted to move to countable sets; and now you want to talk about “non-exclusive” sets. Don’t you realize that your new set is as “exclusive” as any other set? “1 1/4”, for example, cannot be an element of it.

When you add the elements of your new “exclusive” set (“which are the natural numbers plus 1/2”) to the set of natural numbers you are not expanding nor the set of natural numbers nor your new “exclusive” set; you are just defining a third “exclusive” set. Does this third “exclusive” set have more elements than the set of natural numbers? If you can’t establish a bijective relation between it and the set of natural numbers, then I would have to accept that that it does; however, you can establish the relation.

When you read your equations like this:

aleph0+n = aleph0
aleph0*n = aleph0
aleph0^n = aleph0

You might understand that you are not expanding your original set by adding, multiplying, or rising to any power (you are just defining a new “exclusive” set), and that you are confused because you mix your imagination with your reason.

You might continue mentioning your Hilbert’s hotel as many times as you wish. I will ignore it as long as you don’t show your ability to handle it properly.

The point about an infinite universe is that the scale factor can become greater everywhere, and the universe simply remains infinite (you can’t say that it becomes bigger because it’s already infinitely big). There is no logical or mathematical reason why infinite universes with different scale factors cannot exist at different times and so there is no logical or mathematical reason why one scale factor shouldn’t expand into another over time. The expansion over time is finite not infinite and so not excluded logically or mathematically.

Given two infinite sets between which you can establish a bijective relation, how do you determine the “scale factor” of each one? Or the “scale factor” that one of them has over the other.

Once that you have acknowledged that physicists do not know if the universe is finite and connected or infinite, your “mathematical or logical arguments” become irrelevant.

 

[JuanFlorencio]

It’s only under certain mathematical definitions which are particularly relevant for Riemannian manifolds which in turn are particularly relevant to the mathematical theory of General Relativity. Only under the relevant, self-consistent defintions of Riemannian manifolds - look, I have said multiple times that our descriptions in words are misleading - one needs to learn the relevant mathematics to properly understand these points. Wherever I make a statement, I know that it is potentially misleading. You can’t hope to understand GR by words.

No, I said the appearance of acceleration due to gravity is caused by the curvature of spacetime. An object in free-fall undergoing what appears to be acceleration is, in fact, following a geodesic with zero proper acceleration. The notions of curvature are different because in one case its a curvature in space only and in the other case it’s a curvature in spacetime. And indeed the notion of curvature of Riemannian manifolds can only be fully described by the full tensor theory of GR. But I still don’t know what you mean by “generalized notion” of flatness and curvature. Note that in Riemannian geometry the concepts of curvature and flatness are consistent and well-defined.My definition of the angles of a triangle and the behaviour of parallel lines uis a good starting point.

…] you need the maths, and in this case Lie algebras. There are other 3-manifolds including the 3-sphere and other homology spheres such as dodecahedral space which result in compact connected manifiolds.

…] The initiation of the concepts might be promoted by analogies, but the concepts are rigourously and unambiguously defined. Then the analogies inevitably break down.

It is clear to me that when you say that “You can’t hope to understand GR by words” you don’t want to mean that GR is only understood thanks to telepathy or something similar; because obviously your teachers have talked to you. And it is also clear to me that GR is based upon an important number of technical concepts expressed in words whose meaning differ from the common use, or in words which are absent in our common terminology. You have mentioned

• Topology
• Riemann manifolds
• Lie algebras
• Tensor theory

And I think than this could help “I_am_learning” to understand that without those tools, and some others, a discussion about finite or infinite universes is necessarily confusing.

 

[ericc]

And although most of what you wrote is correct, this statement shows a lack of understanding of the physics and the properties of metric spaces:
Not correct:the expansion doesn’t proceed from an initial point; in fact, the initial locus is everywhere in space: cosmic expansion is an increase in the scale factor everywhere. The Big Bang happened everywhere.

That was the part that caused a mental overwork because there was no “where” before the Bang. Before the Bang, nothing. Then a super inflating balloon (universe) suddenly appeared into existence out of nowhere with matter in it, or at least this is is how I imagine it. Resulting in a flat , finite(by the speed of light) but unbounded because it is still expanding but yet possessing a boundary limit at any specific point in time and largely even universe. Would you say that is a fair assessment?

 

[JerryZ]

And although most of what you wrote is correct, this statement shows a lack of understanding of the physics and the properties of metric spaces:
Not correct:the expansion doesn’t proceed from an initial point; in fact, the initial locus is everywhere in space: cosmic expansion is an increase in the scale factor everywhere. The Big Bang happened everywhere.

And there lies your problem, a “singularity” does not have any space!

Nothing is NOT something. The initial locus of the singularity CANNOT be in space. Space did not exist at t = zero

Space expanded inside the locus at t = one trillionth of a trillionth of a trillionth of a second after the Big Bang.

…

 

[hecd2]

You started with the set of real numbers; then you wanted to move to countable sets; and now you want to talk about “non-exclusive” sets. Don’t you realize that your new set is as “exclusive” as any other set? “1 1/4”, for example, cannot be an element of it.

I am just trying to help you understand an aspect of transfinite mathematics that is obviously giving you difficulty. Your position seems to be the bizarre and untenable idea that you cannot add elements to an infinite set to create a new set.

When you add the elements of your new “exclusive” set (“which are the natural numbers plus 1/2”) to the set of natural numbers you are not expanding nor the set of natural numbers nor your new “exclusive” set; you are just defining a third “exclusive” set. Does this third “exclusive” set have more elements than the set of natural numbers?

It has the same cardinality as the set of natural numbers, but, critically, it has more elements on any finite interval > 1 than the original set has. And since an increase in the scale factor of the universe means that any finite distance increases to a greater finite distance then this is completely relevant. In other words, in the same way that you can add elements to an infinite set to create a new infinite set with the same cardinality but a larger number of elements on a finite interval (ie a greater density), so an infinite space can expand by the scale factor increasing locally or globally while the total space remains infinite in extent.

If you can’t establish a bijective relation between it and the set of natural numbers, then I would have to accept that that it does; however, you can establish the relation.

You are looking at the cardinality, I am looking at the density. You can’t say that {1,2,3,4,5…} is the same set as {1, 1.5, 2, 2.5, 3…}. They are diferent sets with different elements and different densities, even though you can establish a one to one relationship between the elements and they have the same cardinality.

You might continue mentioning your Hilbert’s hotel as many times as you wish. I will ignore it as long as you don’t show your ability to handle it properly.

Ignore it if you wish - no skin off my nose. But if you do learn about Hilbert’s Hotel, you will learn that you can arbitrarily add as many elements as you like to any countably infinite set.

Once that you have acknowledged that physicists do not know if the universe is finite and connected or infinite, your “mathematical or logical arguments” become irrelevant.

Of course they don’t. We are not establishing whether the universe is infinite (which of course we cannot do), but whether an infinite universe which is expanding is logically or mathematically excluded. Which it isn’t.

 

[hecd2]

It is clear to me that when you say that “You can’t hope to understand GR by words” you don’t want to mean that GR is only understood thanks to telepathy or something similar; because obviously your teachers have talked to you. And it is also clear to me that GR is based upon an important number of technical concepts expressed in words whose meaning differ from the common use, or in words which are absent in our common terminology. You have mentioned

• Topology
• Riemann manifolds
• Lie algebras
• Tensor theory

And I think than this could help “I_am_learning” to understand that without those tools, and some others, a discussion about finite or infinite universes is necessarily confusing.

Yes that is true, and in addition I am distinguishing between mathematics and the sort of language we are engaged in on this forum, which I have called “words”.

 

[hecd2]

That was the part that caused a mental overwork because there was no “where” before the Bang. Before the Bang, nothing. Then a super inflating balloon (universe) suddenly appeared into existence out of nowhere with matter in it, or at least this is is how I imagine it. Resulting in a flat , finite(by the speed of light) but unbounded because it is still expanding but yet possessing a boundary limit at any specific point in time and largely even universe. Would you say that is a fair assessment?

Up to a point, but not entirely, because what “appeared” could initially have been infinite in extent while being immensely denser and hotter than the current universe.Or it could have been finite and connected. But in neither case do cosmologists consider a boundary as a possible scenario.

 

[hecd2]

And there lies your problem, a “singularity” does not have any space!

I have a problem?

Nothing is NOT something. The initial locus of the singularity CANNOT be in space. Space did not exist at t = zero

OK.

Space expanded inside the locus at t = one trillionth of a trillionth of a trillionth of a second after the Big Bang.

What locus? We just agreed that no space exists before the Big Bang so there can’t be a locus, can there? Where is that locus now?

You still haven’t explained why you think: "“If it [the universe] is expanding then A boundary exists.” is true.

 

[thinkandmull]

The idea of the singularity runs into the same problems as an infinite regress of previous past motions. If there are infinite steps back to the singularity, even though they are progressively getting smaller, what does the singularity as a limit accomplish

 

[thinkandmull]

NOTE: Weiner Heisenberg in his book Physics and Philosophy says that when he considered aspects of infinity in physics contradictions naturally arose

 

[Richca]

We talk about science a lot in my robotics class, and a few things have been talked about which I was not sure about.

1. I hear that more scientific evidence is pointing towards an infinite universe. There was a video that talked about the visible universe, and what we cannot see, and that what we do know about the universe as a whole is that it seems to continue to expand.
   Would this be like saying that the universe is not finite, therefore just an infinite regress, and no God? But isn’t an infinite regress illogical or impossible or something?
2. Some questions about Heaven have been popping up too. The recent one was “can Heaven be in another dimension?” This seems tough to think about, but is that a possibility?

Thanks!

Well, if the universe is expanding, than it is not infinite because if it were infinite it would not be expanding. Similarly, if the universe is expanding, that is, increasing in dimensive quantity or size which means that at any given moment it has a determinate size and is not infinite, it will never become infinite in size even if it expanded for eternity. This is because the infinite is boundless, measureless, limitless.

 

[JuanFlorencio]

I am just trying to help you understand an aspect of transfinite mathematics that is obviously giving you difficulty. Your position seems to be the bizarre and untenable idea that you cannot add elements to an infinite set to create a new set.

Thank you for your good intentions! If you want to help me, there is one way: Don’t use “words”! present a demonstration of what you say, and be careful with your arguments. Isn’t that the best way in mathematics?

What I have to say for the moment is that you have not read my comment correctly: I have said that you don’t add elements to an infinite set (which already contains all possible elements of the set), but that you define a new infinite set. Your description of what you are doing seems inaccurate to me.

It has the same cardinality as the set of natural numbers, but, critically, it has more elements on any finite interval > 1 than the original set has. And since an increase in the scale factor of the universe means that any finite distance increases to a greater finite distance then this is completely relevant. In other words, in the same way that you can add elements to an infinite set to create a new infinite set with the same cardinality but a larger number of elements on a finite interval (ie a greater density), so an infinite space can expand by the scale factor increasing locally or globally while the total space remains infinite in extent.

Ok, now you want to include the notion of density. Then that is fine. It would seem that when establishing the bijective relation between your set and the set of natural numbers, for every finite interval you will leave out some elements of your set, without relation to any natural number. This way, 1 will be related to 1; 2 will be related to 2; 3 will be related to 3, and so on, but 1 1/2 and 2 1/2, will not have a correlate. This is the result of combining your imagination with your reason. It works sometimes, but not when you are dealing with infinite sets. As I suggested above, all you have to do is to develop a proof: define formally your “density” and your “expansion”, and add the appropriate definitions for infinite sets. Then proceed logically and demonstrate your point. It should be easy to you!

When we say that a set S is infinite if it has a proper subset T with which a bijective relation can be established you might think that S is an expansion of T. But in reality S and T satisfy different definitions. I have the impression that this is one of the points where you are confused.

You are looking at the cardinality, I am looking at the density. You can’t say that {1,2,3,4,5…} is the same set as {1, 1.5, 2, 2.5, 3…}. They are diferent sets with different elements and different densities, even though you can establish a one to one relationship between the elements and they have the same cardinality.

Cardinality and density… Yeah, that might be the reason! Now you just have to demonstrate your point in the proper way.

If you realize that natural numbers are not things, but our mental acts of counting, you might understand why your “density” does not clarify anything: no matter what the peculiarities of the elements of a countable set might be, we always count them in the same way: one element, two elements, three elements…, and so on.

Of course they don’t. We are not establishing whether the universe is infinite (which of course we cannot do), but whether an infinite universe which is expanding is logically or mathematically excluded. Which it isn’t.

Which you haven’t demonstrated so far. I hope you can understand that “to say something many times” is not equivalent to “to demonstrate something”.

You had considered previously that the universe would be quantized, so that it would be a countable system. So, if you can demonstrate that, in general, an infinite countable set (remember that you get a set when you define it “exclusively”) can be expanded, I will happily admit that I am wrong.

 

[hecd2]

Thank you for your good intentions! If you want to help me, there is one way: Don’t use “words”! present a demonstration of what you say, and be careful with your arguments. Isn’t that the best way in mathematics?

Yes. You are right that we have both used verbal arguments in this thread which are not rigorously correct; if you remember, that is a point that I have made multiple times. I agree that I have done so in this branch of the thread where we are talking about using the properties of infinite sets to think about an infinite universe, as I have done where we were talking about General Realtivity and cosmology.

So let’s have a couple of definitions, and then I am going to ask you a question and we can go from there.

• An FLRW universe is a homogeneous and isotropic exact solution of the Einstein field equations describing an expanding or contracting universe
• An infinite FLRW universe is flat and is infinite in extent but not necessarily infinite in past time
• For an infinite FLRW universe, the expansion of the universe does not mean that its boundary is expanding since an infinite universe is without boundary
• The expansion of the universe means that the cosmic scale factor changes over time
• Changes in the cosmic scale factor over time results in changes in the relative proper distance of two objects in the universe - (this is called the Hubble flow)
• Changes in distance caused by motion apart from the Hubble flow, arising, for example, from local gravitational interactions, known as peculiar motion, is not included in changes in the scale factor
• The expansion is the same everywhere, so the change in scale factor can be determined by the finite change in the finite proper distance of any two cosmic objects.

The question: given those definitions, do you consider that the concept of an expanding infinite universe can be excluded mathematically or logically? And if so, how would you demonstrate that?

I recognise that you could say that this is a blatant attempt to shift the burden of proof, as it is, but my attempt to use infinite sets in this discussion is not going well. (I think that there might be some mileage in a more rigorous definition of density, and possibly via the Lebesgue measure, but that would need me to learn some measure theory, so that’s not going to be forthcoming in the next day or two).

 

[LogisticsBranch]

Prepared by Dr. William N. Watson, Physicist
DOE [United States Department of Energy] Office of Scientific and Technical Information
Last updated on Thursday 17 December 2015
In the OSTI Collections: Dark Matter and Dark Energy

Recent observations of the universe, combined with Einstein’s theory of general relativity, indicate that most of the universe consists of entities very different from the matter and energy long familiar to us. These previously unknown entities are beginning to be explored on several fronts, many through Department of Energy sponsorship.

Albert Einstein’s theory of relativity describes space and time as observer-dependent aspects of a single absolute entity (spacetime). According to the theory, just as a two-dimensional surface can be curved, four-dimensional spacetime is also curved, with the curvature at different places and times being partly determined by how matter (or equivalently, energy) is distributed within it. Where curvature is lacking, matter will move along straight paths at constant speeds; where spacetime is curved, matter will follow the straightest possible paths along the curves, changing direction and speed as it follows those curved paths. The exact changes described by the theory match the known effects of gravity on the motion of matter. Relativity theory thus represents the mechanism of gravity as the effect of matter curving the spacetime it occupies and then following the straightest possible paths along the curves.

Einstein didn’t simply propose the basic mechanism described above, but worked out a precise equation to relate features of the spacetime curvature to the distribution of matter within spacetime:

[Review online]

. . .]

Observations of more and more distant galaxies during the last century indicated that the universe is indeed getting bigger, its galaxies generally moving farther and farther apart. For most of the time since the first of these observations were made, it was generally expected that the combined effect of matter on the spacetime containing it would be to slow down the expansion of space. But late last century, observations made to determine exactly how much the expansion was decelerating indicated that the expansion was actually accelerating—an impossibility according to Einstein’s equation, unless were larger than a certain value. This can be seen more easily if the equation is algebraically rearranged as follows:

[Review online]

. . .]
osti.gov/home/osti-collections-dark-matter-and-dark-energy
http://www.osti.gov/home/osti-collections-dark-matter-and-dark-energy

I have a tad more to contribute later. Today I’m X-mas shopping! I had a wonder party at my house recently and the doors were open to my dearest of friends that were religious and non-religious. We had a blast. Lot’s of fun and discussing this topic. We all love each other. Happy holidays to everyone!

 

[thinkandmull]

hecd2 seems to be boggled down in the idea that the universe could be infinite yet still expand because all countable infinities are equal anyway. I don’t like Cantor. His diagonal proof is as elegant as a zombie, and I found a great way to refute it today, apart from that the fact that diagonal correspondents would make every infinity in reality equal . Last time I tried people got very mad at me though.