Infinite Universe? Heaven?

[You]

It seems proved that the universe is expanding. It’s infinity is a matter of speculation. It’s topology is a matter of speculation as well. That being said, if for the expanding universe you go mentally in reverse you could go for ever without ever stoping, asymptotically approaching a predetermined minimum size without ever reaching to it. Or you can always conceive a size which is smaller to any other given size, no matter how small it is. Then you would be asymptotically approaching zero without ever reaching to it. But it is because nothing prevents your mind from continuing a dividing operation for ever. However, physical division is a different thing: you cannot go for ever with it. Then it seems to me you are right.

Now, when I imagine the Big Bang, I cannot avoid imagining the initial particle in the middle of space. And if I focus my attention to that space, I can’t avoid thinking that it is infinite, because nothing prevents my mind from going always beyond a certain point (if I look at the particle there is always more space at my back). That happens to me even when I think about a finite universe: there is always room for a expanding universe in my mind. And all that is speculation too.

well, maybe space, the universe and infinity are different things. what if infinity is the normal state outside everything and a finite universe is something that happened in it and the universe expands like a wave outwards through infinity. so the universe is finite but is expanding forever through infinity. instead of trying to call the universe infinite when an expanding universe had a boundary at its beginning and so could be called finite.

 

[hecd2]

Fascinating! It seems to me then that those notions of curvature and flatness are generalized notions; and I wonder if the generalized notion of curvature is incompatible with the generalized notion of flatness.

Not sure what you are saying here - can you clarify?

When you say that large masses curve the space(-time?) locally, does it mean that in the vicinity of these large masses the internal angles of a triangle do not sum up 180 degrees?

Yes, just so. But the curvature in our vicinity is so small as to make triangles indistinguishable from 180 degrees.

Also, what does it mean for the 3-torus to be unbounded? Does it mean that it cannot be “observed” from outside even though it is finite?

It’s unbounded because it has no boundary (just as a 2-torus - which is a 2-D surface has no boundary). In cosmology I don’t know what observation from outside means - there is no “outside” so I don’t think it’s a well-formed question.

I can see that I am missing a number of elements to be able to establish the correct relations…

Unfortuantely, as I pointed out up the thread, talking about this stuff in words is always inaccurate. To understand the “correct relations”, you really need to learn the maths - which is not trivial.

When you see that a body is accelerated as it approaches a large mass, you say (if I have “understood” correctly) that the space is curved there (referring to a generalized notion of curvature). You can look for the large mass which is causing the curvature and you will find it somewhere. On the other hand, if you notice that the universe is expanding more and more rapidly, instead of saying that the space becomes more and more curved there, you prefer to say that there is a dark energy acting like a pressure. I guess these would be generalized notions of energy and pressure, and not the notions that common university people might have. Is it?

So, the term Dark Energy is a place holder for something like the cosmological constant - something is apparently accelerating the expansion of the Universe, and one candidate is a term in the Einstein field equations called the cosmological constant. This would be a form of energy which has a constant density independent of the expansion and an equation of state w=-1 (i.e. it acts like a negative pressure equal to its energy demnsity), sometimes called the energy of the vacuum.

But then, do cosmologists conceive the universe as finite but unbounded because it reconnects?

A finite but unbounded universe would connect (I used the term reconnect because I was doing some other physics at the time where that term is used but the correct word in this context is “connect”).

If so, it seems to me that the reconnection requires an extra dimension which implies a generalized exteriority in respect to the reconnected space. Also, I am very curious now to know which interactions led cosmologists to think that the universe is reconnected and unbounded?

Ah - that’s where analogies break down as well. We use a sphere to explain the concept of a finite but unbounded space, and a sphere is of course, a 2-D surface embedded in a 3-D space. But Gauss proved nearly 200 years ago that curvature of a space (called in maths a manifold) does not require it be embedded in a higher dimension. This sort of curvature is called intrinsic curvature and its is intrinsic curvature that appears in the mathematics of GR - specifically in the maths of non-Euclidean manifolds - Riemannian space. And to yr second question - physicists don’t know if the universe is finite and connected (compact in the jargon) or infinite. Physicists think that the Universe would have to be unbounded because of the physical and philosophical difficulties of a boundary.

There must be a lot of theory behind the interpretation of the CMB phenomena. When you say that you are checking the internal angles of a cosmically large triangle, I cannot figure out what that means. I can imagine that we can draw triangles on the surface of a torus or of a sphere, but you must be speaking of a physical triangle (one that you build somehow in the physical space) not of a mathematical triangle (one that you just conceive in your mind). How do you do that?

It’s as I said - cosmologists measure the angular size of known features in the CMB (specifically the angular scale of the first acoustic peak), in effect checking one angle of an isosceles triangle with equal sides of 13.7 billion light years and a base of known size. If space is flat we know what that angle should be because we know the physical size of the features and therefore what their angular size should be in flat space. As it turns out, the angle measured is very close to that expected for flat space. It’s much more complicated than that in practice, but that’s the principle.

 

[hecd2]

I don’t think the real numbers are a good example to realize if something infinite can expand or not, hecd2: you are not expanding the set of real numbers when you multiply its elements by any finite number, but it remains exactly the same. Besides, numbers are not “there”, but are just mental actions.

You are not multiplying its elements by a finite number, but you are adding one (or some other finite number) element for every existing element.

This is conceptually more difficult to understand with the real numbers. If you look at countably infinite sets, it’s a bit easier conceptually. Take a universe of infinite extent at t0. There are are a countably infinite number of quantised points in the universe. At some later time t1, let’s say that the scale of the universe has expanded by a factor of cube root of 2. Then you have added an uncountably infinite number of quantised points, because you have added one for every existing one, and you are now have a universe which is infinite with an countably infinite number of quantised points - as before. As I said, this is simply a Hilbert’s Hotel.

It is meaningless to talk about the size of an infinite universe, (infinity is not a number), but cosmologists talk about the increase in the scale factor of the universe.

I really don’t see what the conceptual or mathematical objection is to the scale expansion of an infinite universe, ie the scale increasing everywhere.

 

[hecd2]

But precisely a definition of an infinite set is that it has a proper subset with which a bijective function or transformation can be established. Once you have considered the interval between 0 and 1, you can “expand” it to include the interval between 1 and 2, but by doing that you are not expanding the set of real numbers. Whatever interval you consider, it can be associated to an infinity of real numbers, but not all the real numbers will be associated to it: for example, the real numbers associated to the interval between 2 and 3 are not the same as those associated to the interval between 1 and 2. You cannot expand the set of real numbers, and you cannot expand the subset of real numbers associated to any interval.

Yes, but you can add another interval to the subset and you still have a set with an uncountably infinite number of elements, as before.

 

[hecd2]

but if the universe is expanding then you could draw lines backwards to a point which cannot logically become infinitely small. There is a limit on infinity in an expanding universe.

If you start with the assumption of an infinite universe and look back in time what is the size of the universe as the density approaches infinity? Well, it’s still infinite. And when the density becomes actually infinite - the size is undefined mathematically, as is the density, because we don’t actually know what an infinite density means physically. Physics starts after the instant of the singularity when the Universe would have been immensely hot and dense - and could have been infinite in extent. The scale factor would have increased by a factor 10^50 since then and the universe would still be infinite.

 

[You]

If you start with the assumption of an infinite universe and look back in time what is the size of the universe as the density approaches infinity? Well, it’s still infinite. And when the density becomes actually infinite - the size is undefined mathematically, as is the density, because we don’t actually know what an infinite density means physically. Physics starts after the instant of the singularity when the Universe would have been immensely hot and dense - and could have been infinite in extent. The scale factor would have increased by a factor 10^50 since then and the universe would still be infinite.

if we work backwards the distance between me and the nearest star decreases, that is the distance stops approaching infinity and moves backwards away from infinity.
the mass of the universe may be constant so it may possibly in theory become infinitely dense but its volume doesn’t remain constant it gets smaller the further back you travel and larger the further forwards to infinity you travel.

 

[JuanFlorencio]

You are not multiplying its elements by a finite number, but you are adding one (or some other finite number) element for every existing element.

This is conceptually more difficult to understand with the real numbers. If you look at countably infinite sets, it’s a bit easier conceptually. Take a universe of infinite extent at t0. There are are a countably infinite number of quantised points in the universe. At some later time t1, let’s say that the scale of the universe has expanded by a factor of cube root of 2. Then you have added an uncountably infinite number of quantised points, because you have added one for every existing one, and you are now have a universe which is infinite with an countably infinite number of quantised points - as before. As I said, this is simply a Hilbert’s Hotel.

It is meaningless to talk about the size of an infinite universe, (infinity is not a number), but cosmologists talk about the increase in the scale factor of the universe.

I really don’t see what the conceptual or mathematical objection is to the scale expansion of an infinite universe, ie the scale increasing everywhere.

It is fine if you want to consider a countably infinite set now. That system would be associated to the set of natural numbers. Does the set of natural numbers “expand” or “grow” in such a way that each time it comprehends more and more natural numbers? No, it does not “grow”; you cannot add more elements to it. Then, when you imagine the infinite universe expanding and still keep countable, inadvertently you associate to it a growing set of natural numbers, which cannot be. Your imagination comes into conflict with your mathematical reasoning.

But there is no conflict when conceiving a finite expanding universe with quantized points. If it is finite, it can expand; if it is infinite, it cannot.

 

[hecd2]

It is fine if you want to consider a countably infinite set now. That system would be associated to the set of natural numbers. Does the set of natural numbers “expand” or “grow” in such a way that each time it comprehends more and more natural numbers? No, it does not “grow”; you cannot add more elements to it. Then, when you imagine the infinite universe expanding and still keep countable, inadvertently you associate to it a growing set of natural numbers, which cannot be. Your imagination comes into conflict with your mathematical reasoning.

But there is no conflict when conceiving a finite expanding universe with quantized points. If it is finite, it can expand; if it is infinite, it cannot.

I’m afraid you are wrong both logically and mathematically. You can always add elements to an infinite set. As I said - it’s a Hilbert Hotel.

 

[JuanFlorencio]

I’m afraid you are wrong both logically and mathematically. You can always add elements to an infinite set. As I said - it’s a Hilbert Hotel.

Which new elements can you add to the set of natural numbers? I assume as many as you want; but please mention just three of them.

 

[JuanFlorencio]

Not sure what you are saying here - can you clarify?

You were saying that “It comes a surprise to many people that the surface of a torus is flat but it is, because it satisfies the definition of a flat surface”; and I think it was no less surprising to you when you heard about it for the first time. And the surprise originates from our tendency to distinguish between flat and curved surfaces based on the common usage of terms. We would say that the surface of a torus is obviously curved; but when you characterize a flat surface mathematically it appears that the surface of the torus falls under that characterization. Instead of thinking that your characterization of the flat surface was not good enough because it encloses surfaces which are evidently not flat, you prefer to say: well, surprisingly the surface that we thought curved is flat! Then I say that you have adopted a generalized notion of flatness. The notion of acceleration has nothing to do with the definition of flatness (as far as I can see from your words).

Regarding curvature: you say that the acceleration of a body approaching a large mass is caused by a curvature of the space in the vicinity of the mass; and that any large mass curves the space in its vicinity. Such curvature is not perceived by us as we perceive the curvature of a sphere. So, I say that you are using a generalized notion of curvature. I don’t see any relation between this notion of curvature (curvature I) and the curvature that is opposed to the generalized notion of flatness (curvature II).

Then I say: “I wonder if the generalized notion of curvature (I) is incompatible with the generalized notion of flatness.”

It’s unbounded because it has no boundary (just as a 2-torus - which is a 2-D surface has no boundary). In cosmology I don’t know what observation from outside means - there is no “outside” so I don’t think it’s a well-formed question.

I think that either for a cosmologist or for a non-cosmologist the idea that the universe has no boundary is the result of a thought process: A thought process with which you end thinking that the universe topology is analogous to a torus. And in a topology class you can present to me a torus which I will look at from outside. In the same manner, when you think on the universe as something analogous to a torus, nothing prevents you from thinking of an “outside” the shape.

Unfortuantely, as I pointed out up the thread, talking about this stuff in words is always inaccurate. To understand the “correct relations”, you really need to learn the maths - which is not trivial. So, the term Dark Energy is a place holder for something like the cosmological constant - something is apparently accelerating the expansion of the Universe, and one candidate is a term in the Einstein field equations called the cosmological constant. This would be a form of energy which has a constant density independent of the expansion and an equation of state w=-1 (i.e. it acts like a negative pressure equal to its energy demnsity), sometimes called the energy of the vacuum.
A finite but unbounded universe would connect (I used the term reconnect because I was doing some other physics at the time where that term is used but the correct word in this context is “connect”).

It makes a lot of sense to me that scientists look for analogies to explain new phenomena. I just wanted to point out that those notions of “pressure” (positive or negative, whatever) and “dark energy”, are the result of analogical thinking. Analogies do not break down. You just have to look for the good one.

Ah - that’s where analogies break down as well. We use a sphere to explain the concept of a finite but unbounded space, and a sphere is of course, a 2-D surface embedded in a 3-D space. But Gauss proved nearly 200 years ago that curvature of a space (called in maths a manifold) does not require it be embedded in a higher dimension. This sort of curvature is called intrinsic curvature and its is intrinsic curvature that appears in the mathematics of GR - specifically in the maths of non-Euclidean manifolds - Riemannian space. And to yr second question - physicists don’t know if the universe is finite and connected (compact in the jargon) or infinite. Physicists think that the Universe would have to be unbounded because of the physical and philosophical difficulties of a boundary.

Yes, no doubt I need to know Gauss’ proof. I am working on that this years.

And is this intrinsic curvature a third notion, or somehow the three notions that you have mentioned are three aspects of the same curvature?

Thanks for your response. If you have the authority to say that physicists don’t know if the universe is finite or not, then you can tell “I_am_learning” where he can look for the details of the speculations.

It’s as I said - cosmologists measure the angular size of known features in the CMB (specifically the angular scale of the first acoustic peak), in effect checking one angle of an isosceles triangle with equal sides of 13.7 billion light years and a base of known size. If space is flat we know what that angle should be because we know the physical size of the features and therefore what their angular size should be in flat space. As it turns out, the angle measured is very close to that expected for flat space. It’s much more complicated than that in practice, but that’s the principle.

I think I can understand about the difficulties of the determination. Sometimes, measurements that would seem simple to us, are in reality quite complex.

 

[yppop]

The way I deal with this whole business of the creation of reality is as follows:

1. The rational numbers, the natural numbers plus their ratios, are infinite as represented by the first transfinite number, aleph(0). They are countable because there are gaps, so each rational number has a next neighbor.
2. The real numbers are all the rational numbers plus the irrational numbers. The irrationals (like pi), are decimals that never end and are described by the second transfinite number aleph(1). The irrationals are infinitely divisible, can’t be counted because there are no next number.
3. Since Cantor proved that every point in 3-dimensional space can be associated with a single number on the real number line, the two sets of numbers, rational and real can be used to represent two kinds of space. The rational number represent the points of discrete space; the real numbers represent the points of continuous space.
4. Since the real numbers subsume the rational numbers, continuous space subsumes discrete space. Hence we can begin with an infinite extent of continuous space in which a finite number of discrete points are immersed then caused to expand to become the universe.
5. Such a scenario prefaces a finite cosmic configuration of discrete points immersed in an infinite ocean of continuous points. When matter is formed from localized configuration of discrete points, a hylomorphic structure of material and spiritual, results and evolves into what we experience as reality.

This is a very abbreviated description of a much more extensive argument for the existence of God, not a proof, but a plausible explanation of how God might involve in our reality.

Yppop

 

[JuanFlorencio]

1. The real numbers are all the rational numbers plus the irrational numbers. The irrationals (like pi), are decimals that never end and are described by the second transfinite number aleph(1). The irrationals are infinitely divisible, can’t be counted because there are no next number.

When you want to approximate the value of an irrational number you do it by means of rational numbers, with more and more decimals as you want to have a better approximation to them. Then, those representations that you see with many decimals are not the representation of irrational but rational numbers (possibly close to an irrational number), based on the decimal system of symbols and construction rules.

It is necessary to distinguish between numbers and the symbols we use to represent them.

 

[You]

i don’t understand maths. the universe doesn’t understand maths. a butterfly is not maths. maths approximate stuff but its never really absolute or is it?

 

[hecd2]

Which new elements can you add to the set of natural numbers? I assume as many as you want; but please mention just three of them.

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.

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.

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.

 

[JerryZ]

I don’t see why not given that the cardinality of the real numbers on any finite number interval is the same (the cardinality of the continuum). Or to put it another way, n times infinity equals infinity where n is any finite number.

That’s a nice answer where it not we live in actual real space. Space cannot be “infinite” and “expanding” at the same time.

If it is “Infinite” it does not, IT CANNOT have a boundary.
If it is expanding then A boundary exists.

To say that the Universe is infinite AND expanding is a non logical statement.

…

 

[hecd2]

You were saying that “It comes a surprise to many people that the surface of a torus is flat but it is, because it satisfies the definition of a flat surface”; and I think it was no less surprising to you when you heard about it for the first time.

Yes it was.

And the surprise originates from our tendency to distinguish between flat and curved surfaces based on the common usage of terms. We would say that the surface of a torus is obviously curved; but when you characterize a flat surface mathematically it appears that the surface of the torus falls under that characterization.

Yes. 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.

Instead of thinking that your characterization of the flat surface was not good enough because it encloses surfaces which are evidently not flat, you prefer to say: well, surprisingly the surface that we thought curved is flat

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.

Then I say that you have adopted a generalized notion of flatness. The notion of acceleration has nothing to do with the definition of flatness (as far as I can see from your words).

I still don’t know what you mean by a “generalized notion of flatness”. And there are different sorts of acceleration so there is no one definition.

Regarding curvature: you say that the acceleration of a body approaching a large mass is caused by a curvature of the space in the vicinity of the mass;

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.

and that any large mass curves the space in its vicinity. Such curvature is not perceived by us as we perceive the curvature of a sphere. So, I say that you are using a generalized notion of curvature. I don’t see any relation between this notion of curvature (curvature I) and the curvature that is opposed to the generalized notion of flatness (curvature II).

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.

Then I say: “I wonder if the generalized notion of curvature (I) is incompatible with the generalized notion of flatness.”

See above.

I think that either for a cosmologist or for a non-cosmologist the idea that the universe has no boundary is the result of a thought process: A thought process with which you end thinking that the universe topology is analogous to a torus

Not necessarily - again 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.

And in a topology class you can present to me a torus which I will look at from outside. In the same manner, when you think on the universe as something analogous to a torus, nothing prevents you from thinking of an “outside” the shape.

If you can think of a 3-torus from “outside the shape” then you’re a better man than I am, Gunga Din. But in any case, the curvature of Riemannian manifolds are intrinsic and do not need to be embedded in higher dimensions.

It makes a lot of sense to me that scientists look for analogies to explain new phenomena. I just wanted to point out that those notions of “pressure” (positive or negative, whatever) and “dark energy”, are the result of analogical thinking. Analogies do not break down. You just have to look for the good one.

This is not so. The initiation of the concepts might be promoted by analogies, but the concepts are rigourously and unambiguously defined. Then the analogies inevitably break down.

And is this intrinsic curvature a third notion, or somehow the three notions that you have mentioned are three aspects of the same curvature?

What three notions? You need to learn the maths. Words won’t suffice.

 

[hecd2]

That’s a nice answer where it not we live in actual real space. Space cannot be “infinite” and “expanding” at the same time.

If it is “Infinite” it does not, IT CANNOT have a boundary.
If it is expanding then A boundary exists.

To say that the Universe is infinite AND expanding is a non logical statement.

…

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

 

[hecd2]

The way I deal with this whole business of the creation of reality is as follows:

1. The rational numbers, the natural numbers plus their ratios, are infinite as represented by the first transfinite number, aleph(0). They are countable because there are gaps, so each rational number has a next neighbor.
2. The real numbers are all the rational numbers plus the irrational numbers. The irrationals (like pi), are decimals that never end and are described by the second transfinite number aleph(1). The irrationals are infinitely divisible, can’t be counted because there are no next number.
3. Since Cantor proved that every point in 3-dimensional space can be associated with a single number on the real number line, the two sets of numbers, rational and real can be used to represent two kinds of space. The rational number represent the points of discrete space; the real numbers represent the points of continuous space.
4. Since the real numbers subsume the rational numbers, continuous space subsumes discrete space. Hence we can begin with an infinite extent of continuous space in which a finite number of discrete points are immersed then caused to expand to become the universe.
5. Such a scenario prefaces a finite cosmic configuration of discrete points immersed in an infinite ocean of continuous points. When matter is formed from localized configuration of discrete points, a hylomorphic structure of material and spiritual, results and evolves into what we experience as reality.

This is a very abbreviated description of a much more extensive argument for the existence of God, not a proof, but a plausible explanation of how God might involve in our reality.

Yppop

No problem with 1 to 3

Why not:
4. Since the real numbers subsume the rational numbers, continuous space subsumes discrete space. Hence we can begin with an infinite extent of continuous space in which an infinite number of discrete points are immersed then caused to expand to become the universe.

1. Such a scenario prefaces an infinite cosmic configuration of discrete points immersed in an infinite ocean of continuous points.

And what evidence is there for the idea that space consists of discrete points in an underlying continuum?

 

[thinkandmull]

I am Learning 2

If the universe started out as an infinitesimal object called the singularity, exploded and at one point in its evolution was the size of a pumpkin, it was finite then and it must still be finite. Nothing that is finite can ever reach infinity because by its very definition, infinity can never be reached. A finite universe implies a beginning and a beginning implies the existence of God. To avoid this, materialistic scientists look for a way around the Big Bang implication of a beginning.

If the universe is finite, totality (both known and unknown reality) is divided into two realms separated by a boundary. If we were travel to the end of the universe, we would run into an impenetrable barrier, simply because we cannot travel beyond the universe. Many cosmologists deny such a boundary exists and argue as Einstein did that the universe is finite but unbounded. According to the General Theory of Relativity, space-time bends in the presence of large masses. Therefore, the total mass of the universe bends space such that a light wave would bend back in upon its starting point and would never reach a boundary. This argument applies only to space with positive curvature. However, recent data indicates that the space has a negative curvature and its expansion is slightly accelerating. Therefore, we have something of a contradiction here. Einstein’s finite and unbounded universe doesn’t seem to be an option. In addition, when cosmologists calculated the cosmic microwave background, they assumed a black body condition for which the radiation was contained within the universe by reflection from a boundary. Those calculations have been verified by observation of the cosmic microwave background, which suggests that there is a boundary between our universe and what came before and still lies beyond. Thus we can argue that the finiteness of the universe, the flatness of universal space, and the observation of the cosmic microwave background implies that the universe has a boundary. How else can a universe the size of a pumpkin, whose space is not curved in upon itself not have a boundary? It must and I argue that the universe is finite and bounded, the only scenario that is both simple and easily imagined.

I suspect that those that can’t live with a finite, bounded universe don’t want to think about the boundary and the obvious question it raises—what is the nature of such a boundary and what came before and lies beyond the universe—because the obvious answer is God.

In thinking about a boundary that separates our universe from the before/beyond, I imagine that at the edge of the universe, like a minimalist painting of black on black, there is a finely delineated perception of two forms of blackness: one form of blackness is the space of our universe; the other form of blackness is the infinite nothingness, the only thing I can imagine the before/beyond to be. According to the big bang theory, expanding space defines the volume of our universe. Therefore, like the skin of a bubble, the imperceptible delineation that defines the limits of the edge of the expansion, divides total reality into two realms: an expandable and bendable space on one side and an unchanging, infinite nothingness on the other. This can only mean that total reality is bifurcated by two kinds of space.

The space that defines the dimensions of the universe must be discrete ( can be described by the rational numbers) and the space that existed before and lies beyond the universe must be continuous (can be described by the real numbers). Based on this structure we can argue that discrete space is the material substance from which matter is formed and continuous space is the immaterial substance which describes the omnipresent psychical/spiritual substance that matter is hylomorphically immersed in.

Heaven is another matter.
Yppop

That was a great post think you are mistaken that matter in our world can be discrete. Anything that has mass will be capable of being divided. I would think the discreet would apply more to heaven than earth. Could you explain what you mean by “the flatness of universal space” and “negative curvature”?

 

[thinkandmull]

yppop writes below in an authoritative tone. But much of what he writes is wrong. There are far more more incorrect statements than I can deal with in a single post so I will just select a few. The point is that yppop’s post is not authoritative and his misunderstanding of the science undermines much of what he has to say. There is no such hypothesis of a boundary in cosmology, so this is ypop’s unsupported notion. The idea that a boundary exists is excluded for physical and philosophical reasons. I’m not aware of a single professional cosmologist who thinks that a boundary is a serious hypothesis. FTFY No - all measurements indicate that space is FLAT although it is also true the expansion is accelerating Einstein’s GR doesn’t favour an unbounded finite universe. In GR, the universe might be finite or infinite, flat, positively or negatively curved but it is generally taken to be unbounded in all cases. This is absolutely not true. There is no assumption in the theory of the CMB that the “radiation is contained by reflection from a boundary”. Where on earth does yppop get that idea? . No, just no. That’s simply not true. You can argue no such thing. First, flatness plus a finite universe do not mandate a boundary. A toroidal topology is flat and finite, and unbounded. Furthermore nothing about the CMB is evidence for a boundary. But there is no evidence that the universe was ever the size of a pumpkin. There is evidence that the OBSERVABLE universe was once very small, but that is an entirely different claim - the observable universe could have been the size of a pumpkin and yet the universe could have been then, and would be now, infinite.

Or it could be finite and unbounded. What is unlikely according to serious cosmological hypotheses is a boundary.

I liked yppop’s post. What philosophical argument are you referring to that proves there is no boundary to our universe?