CaptainToonces

2010-May-18, 08:07 AM

If the universe is finite, then it must have a center of gravity. In which direction is it?

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CaptainToonces

2010-May-18, 08:07 AM

If the universe is finite, then it must have a center of gravity. In which direction is it?

Jens

2010-May-18, 08:34 AM

It doesn't have a center of gravity, as far as we know. From our point of view, we seem to be in the center of the universe. But we presume that this would be true for anybody, i.e. that we are not in a privileged position.

01101001

2010-May-18, 12:44 PM

If the universe is finite, then it must have a center of gravity.

Does not follow from the assumption.

Does not follow from the assumption.

Shaula

2010-May-18, 04:14 PM

Think of Pacman space - when you go off the side of the screen you reappear on the other one. In a spacetime like that (which is perfectly possible) there is no centre of mass. You can only define one by imposing artificial boundaries on which particular way to dice the repeating volume - so the CoM is where you put it!

Jeff Root

2010-May-18, 07:08 PM

If the universe is finite, then it must have a center of gravity.

Does not follow from the assumption.

It does follow if you don't realize that space might not be

Euclidean.

-- Jeff, in Minneapolis

Does not follow from the assumption.

It does follow if you don't realize that space might not be

Euclidean.

-- Jeff, in Minneapolis

swampyankee

2010-May-18, 07:57 PM

GR would also break if there was a definable center of mass for the Universe. At least that's what I think: a definite c/m means there is a point from which an absolute coordinate system could be defined.

astromark

2010-May-18, 08:16 PM

The only way to answer this question is... Not to answer it at all. 'IF' the universe is finite 'AND' was born from a single point. Then it should be possible to pinpoint a center of mass. As we seem to be unable to do that because of eccelorating expansion and shear size....

Then if we can not find such center of mass there might not be one and all bets are off.

Regarding the singular Big Bang of initial creation...

I can except that there are some things that we may never actually know. The deeper you dig into this. The harder to find is a answer.

If you consider that as time is part of the universe and thus did not exist pre universe. Time and space are part of the whole thing so finding a start point or time is never possible... Where is the center of the Universe ? ... Everywhere.

Then if we can not find such center of mass there might not be one and all bets are off.

Regarding the singular Big Bang of initial creation...

I can except that there are some things that we may never actually know. The deeper you dig into this. The harder to find is a answer.

If you consider that as time is part of the universe and thus did not exist pre universe. Time and space are part of the whole thing so finding a start point or time is never possible... Where is the center of the Universe ? ... Everywhere.

Van Rijn

2010-May-18, 08:37 PM

The only way to answer this question is... Not to answer it at all. 'IF' the universe is finite 'AND' was born from a single point. Then it should be possible to pinpoint a center of mass.

Why should it be possible? You seem to be thinking of the misconception of the Big Bang as an explosion in space.

From here:

http://www.xs4all.nl/~johanw/PhysFAQ/Relativity/GR/centre.html

There is no centre of the universe! According to the standard theories of cosmology, the universe started with a "Big Bang" about 14 thousand million years ago and has been expanding ever since. Yet there is no centre to the expansion; it is the same everywhere. The Big Bang should not be visualised as an ordinary explosion. The universe is not expanding out from a centre into space; rather, the whole universe is expanding and it is doing so equally at all places, as far as we can tell.

Why should it be possible? You seem to be thinking of the misconception of the Big Bang as an explosion in space.

From here:

http://www.xs4all.nl/~johanw/PhysFAQ/Relativity/GR/centre.html

There is no centre of the universe! According to the standard theories of cosmology, the universe started with a "Big Bang" about 14 thousand million years ago and has been expanding ever since. Yet there is no centre to the expansion; it is the same everywhere. The Big Bang should not be visualised as an ordinary explosion. The universe is not expanding out from a centre into space; rather, the whole universe is expanding and it is doing so equally at all places, as far as we can tell.

DrRocket

2010-May-18, 08:41 PM

If the universe is finite, then it must have a center of gravity. In which direction is it?

Even if the universe is finite, and there is no proof that such is the case, it does not follow that it has a center of gravity.

Think about an irregularly shaped body, or even a hollow sphere. The center of gravity does not lie on the body, but rather it is a point in the space in which the body is embedded.

So far as anyone knows the universe is an intrincic manifold and it is not embedded in anything.

Even if the universe is finite, and there is no proof that such is the case, it does not follow that it has a center of gravity.

Think about an irregularly shaped body, or even a hollow sphere. The center of gravity does not lie on the body, but rather it is a point in the space in which the body is embedded.

So far as anyone knows the universe is an intrincic manifold and it is not embedded in anything.

DrRocket

2010-May-18, 09:10 PM

The only way to answer this question is... Not to answer it at all. 'IF' the universe is finite 'AND' was born from a single point. Then it should be possible to pinpoint a center of mass.

Wrong. In fact it would be difficult to be more wrong on so many points.

If the universe is finite and was born from a single point, then everywhere in space is the point of origin "now". There still may be no center of mass, depending on the topology of the universe. And if there is it might even be the case that, under the assumption of homogeneity, every point is the center of mass.

On the other hand if, as you seem to be suggesting one looks at the entire spacetime manifold, and one assumes that it began as a point (and the singularity predicted is not just some simple point but we'll make your assumption anyway), then that point is the point of origin of both space and time. However, the way the question is usually framed is in terms of some point in a space-like slice of spacetime which is a different kettle of fish. see above.

Wrong. In fact it would be difficult to be more wrong on so many points.

If the universe is finite and was born from a single point, then everywhere in space is the point of origin "now". There still may be no center of mass, depending on the topology of the universe. And if there is it might even be the case that, under the assumption of homogeneity, every point is the center of mass.

On the other hand if, as you seem to be suggesting one looks at the entire spacetime manifold, and one assumes that it began as a point (and the singularity predicted is not just some simple point but we'll make your assumption anyway), then that point is the point of origin of both space and time. However, the way the question is usually framed is in terms of some point in a space-like slice of spacetime which is a different kettle of fish. see above.

astromark

2010-May-18, 10:04 PM

What is wrong with your understanding Mark ? much, these people would be telling me... and they not me are in error.

... Did they see the 'IF' and the 'AND'... No they did not., or if they did they ignored it... which is a shame because I agree with them. That is what I have said. There is NO center to be found. Its all of it.

Energy became matter and is and has been expanding ever since. I used the word everywhere.... and am right to do so. Read my post again.

... Did they see the 'IF' and the 'AND'... No they did not., or if they did they ignored it... which is a shame because I agree with them. That is what I have said. There is NO center to be found. Its all of it.

Energy became matter and is and has been expanding ever since. I used the word everywhere.... and am right to do so. Read my post again.

DrRocket

2010-May-18, 10:07 PM

What is wrong with your understanding Mark ? much, these people would be telling me... and they not me are in error.

... Did they see the 'IF' and the 'AND'... No they did not., or if they did they ignored it... which is a shame because I agree with them. That is what I have said. There is NO center to be found. Its all of it.

Energy became matter and is and has been expanding ever since. I used the word everywhere.... and am right to do so. Read my post again.

I read it, and my response stands.

... Did they see the 'IF' and the 'AND'... No they did not., or if they did they ignored it... which is a shame because I agree with them. That is what I have said. There is NO center to be found. Its all of it.

Energy became matter and is and has been expanding ever since. I used the word everywhere.... and am right to do so. Read my post again.

I read it, and my response stands.

Jeff Root

2010-May-18, 10:43 PM

GR would also break if there was a definable center of mass for

the Universe. At least that's what I think: a definite c/m means

there is a point from which an absolute coordinate system could

be defined.

The Universe having a center would not break general relativity.

General relativity shows that it is possible to accurately describe

the Universe without a center. But general relativity does not

require that the Universe has no center. The Earth has a center,

and that doesn't break GR. The Milky Way has a center, and that

doesn't break GR. The Universe could have a center, too, without

breaking GR. A center just isn't isn't seen, and isn't required by

what is seen.

-- Jeff, in Minneapolis

the Universe. At least that's what I think: a definite c/m means

there is a point from which an absolute coordinate system could

be defined.

The Universe having a center would not break general relativity.

General relativity shows that it is possible to accurately describe

the Universe without a center. But general relativity does not

require that the Universe has no center. The Earth has a center,

and that doesn't break GR. The Milky Way has a center, and that

doesn't break GR. The Universe could have a center, too, without

breaking GR. A center just isn't isn't seen, and isn't required by

what is seen.

-- Jeff, in Minneapolis

Jeff Root

2010-May-18, 11:22 PM

'IF' the universe is finite 'AND' was born from a single point.

Then it should be possible to pinpoint a center of mass.

Why should it be possible? You seem to be thinking of the

misconception of the Big Bang as an explosion in space.

From here:

http://www.xs4all.nl/~johanw/PhysFAQ...GR/centre.html

There is no centre of the universe! According to the standard theories of

cosmology, the universe started with a "Big Bang" about 14 thousand million

years ago and has been expanding ever since. Yet there is no centre to the

expansion; it is the same everywhere. The Big Bang should not be visualised

as an ordinary explosion. The universe is not expanding out from a centre

into space; rather, the whole universe is expanding and it is doing so equally

at all places, as far as we can tell.

I will argue with the quote. The expansion can be accurately described in

terms of Reimannian geometry as centerless, but it might be described just

as accurately as like an explosion in Euclidean space. That would require

some unknown physics, though. The unknown physics has been shown to

actually exist: It is the acceleration of the expansion, discovered 12 years

ago. Without the unknown physics, Reimannian geometry is needed. With

the unknown physics, Euclidean geometry may suffice. However, since we

don't see any indication of a center or an edge, a centerless Universe is still

the preferred description.

There is no reason the Universe -- or the part of the Universe that came out

of the Big Bang and is still participating in the expansion -- couldn't have a

center. If it has a center, evidence of it may or may not be visible to us. But

the evidence so far doesn't hint at the existence of a center or its location.

-- Jeff, in Minneapolis

Then it should be possible to pinpoint a center of mass.

Why should it be possible? You seem to be thinking of the

misconception of the Big Bang as an explosion in space.

From here:

http://www.xs4all.nl/~johanw/PhysFAQ...GR/centre.html

There is no centre of the universe! According to the standard theories of

cosmology, the universe started with a "Big Bang" about 14 thousand million

years ago and has been expanding ever since. Yet there is no centre to the

expansion; it is the same everywhere. The Big Bang should not be visualised

as an ordinary explosion. The universe is not expanding out from a centre

into space; rather, the whole universe is expanding and it is doing so equally

at all places, as far as we can tell.

I will argue with the quote. The expansion can be accurately described in

terms of Reimannian geometry as centerless, but it might be described just

as accurately as like an explosion in Euclidean space. That would require

some unknown physics, though. The unknown physics has been shown to

actually exist: It is the acceleration of the expansion, discovered 12 years

ago. Without the unknown physics, Reimannian geometry is needed. With

the unknown physics, Euclidean geometry may suffice. However, since we

don't see any indication of a center or an edge, a centerless Universe is still

the preferred description.

There is no reason the Universe -- or the part of the Universe that came out

of the Big Bang and is still participating in the expansion -- couldn't have a

center. If it has a center, evidence of it may or may not be visible to us. But

the evidence so far doesn't hint at the existence of a center or its location.

-- Jeff, in Minneapolis

undidly

2010-May-18, 11:50 PM

If the universe is finite, then it must have a center of gravity. In which direction is it?

If you are on the surface of Earth then it is obvious that the center of the Earth is at right angles to the two directions of movement on the surface.

To explain more fully I must be in ATM.

If you are on the surface of Earth then it is obvious that the center of the Earth is at right angles to the two directions of movement on the surface.

To explain more fully I must be in ATM.

astromark

2010-May-19, 04:06 AM

I want so much for you to understand me... looking at the whole posting i have quoted below. I HAVE HIGHLIGHTED THE SENTENCE YOU SEEM TO HAVE NOT SEEN....

The only way to answer this question is... Not to answer it at all. 'IF' the universe is finite 'AND' was born from a single point. Then it should be possible to pinpoint a center of mass. As we seem to be unable to do that because of eccelorating expansion and shear size....

* ''' Then if we can not find such center of mass there might not be one and all bets are off.''' *

Regarding the singular Big Bang of initial creation...

I can except that there are some things that we may never actually know. The deeper you dig into this. The harder to find is a answer.

If you consider that as time is part of the universe and thus did not exist pre universe. Time and space are part of the whole thing so finding a start point or time is never possible... Where is the center of the Universe ? ... * Everywhere.*

The only way to answer this question is... Not to answer it at all. 'IF' the universe is finite 'AND' was born from a single point. Then it should be possible to pinpoint a center of mass. As we seem to be unable to do that because of eccelorating expansion and shear size....

* ''' Then if we can not find such center of mass there might not be one and all bets are off.''' *

Regarding the singular Big Bang of initial creation...

I can except that there are some things that we may never actually know. The deeper you dig into this. The harder to find is a answer.

If you consider that as time is part of the universe and thus did not exist pre universe. Time and space are part of the whole thing so finding a start point or time is never possible... Where is the center of the Universe ? ... * Everywhere.*

Aik4on

2010-May-19, 06:51 AM

In reply to the OP, if the universe is and always has been 'everything' then, from an observer's point of view, the direction in which the centre of universe lies is always 'everywhere'.

It's not profitable to visualise the universe as a 3D shape with a 'centre' as this implies it is expanding in to some form of higher dimensional space. Saying that, it's very hard not to visualise it in this way as our perception and reality is based on observing 3D objects and the 'space' around them.

It's not profitable to visualise the universe as a 3D shape with a 'centre' as this implies it is expanding in to some form of higher dimensional space. Saying that, it's very hard not to visualise it in this way as our perception and reality is based on observing 3D objects and the 'space' around them.

CaptainToonces

2010-May-19, 07:59 AM

Thanks to Jeff Root for the only sensible answer of them all :)

What a lot of you seem to be overlooking, barring perhaps the introduction of extra dimensions, is that the word "expands" itself implies a center, because expands means to gain volume, and anything with volume has a center, so you cannot say that the universe is expanding without saying it has a center. I think the correct answer to my question, is really, "we don't know."

What a lot of you seem to be overlooking, barring perhaps the introduction of extra dimensions, is that the word "expands" itself implies a center, because expands means to gain volume, and anything with volume has a center, so you cannot say that the universe is expanding without saying it has a center. I think the correct answer to my question, is really, "we don't know."

CaptainToonces

2010-May-19, 08:05 AM

Think of Pacman space - when you go off the side of the screen you reappear on the other one. In a spacetime like that (which is perfectly possible) there is no centre of mass. You can only define one by imposing artificial boundaries on which particular way to dice the repeating volume - so the CoM is where you put it!

Pacman space is simply a torus. Or since there are no passages in Pacman that go up/down, only left/right ones, Pacman is a cylinder. If you could have up/down passages too, it would be a torus. Both these things do have centers. In fact it is impossible to define a 3d shape with volume that does not have a center.

Pacman space is simply a torus. Or since there are no passages in Pacman that go up/down, only left/right ones, Pacman is a cylinder. If you could have up/down passages too, it would be a torus. Both these things do have centers. In fact it is impossible to define a 3d shape with volume that does not have a center.

Shaula

2010-May-19, 08:07 AM

What a lot of you seem to be overlooking, barring perhaps the introduction of extra dimensions, is that the word "expands" itself implies a center, because expands means to gain volume, and anything with volume has a center, so you cannot say that the universe is expanding without saying it has a center.

That is wrong. All expansion means is that if you take two points in space then over time they end up further apart as space itself has expanded. That can happen in an infinite universe, in a finite but unbounded universe and so on. As for the centre - why assume it is inside the shape it is associated with? As has been said - a torus has a centre but that centre is not within the torus. So if the universe were that shape then the 'centre' of it would be outside the universe.

That is wrong. All expansion means is that if you take two points in space then over time they end up further apart as space itself has expanded. That can happen in an infinite universe, in a finite but unbounded universe and so on. As for the centre - why assume it is inside the shape it is associated with? As has been said - a torus has a centre but that centre is not within the torus. So if the universe were that shape then the 'centre' of it would be outside the universe.

Shaula

2010-May-19, 08:11 AM

Pacman space is simply a torus. Or since there are no passages in Pacman that go up/down, only left/right ones, Pacman is a cylinder. If you could have up/down passages too it would be a torus. Both these things do have centers. In fact it is impossible to define a 3d shape with volume that does not have a center.

No, Pacman space is 2D. It is flat. So it is a simple tiled space.

No, Pacman space is 2D. It is flat. So it is a simple tiled space.

CaptainToonces

2010-May-19, 08:16 AM

That is wrong. All expansion means is that if you take two points in space then over time they end up further apart as space itself has expanded. That can happen in an infinite universe

No, that cannot happen uniformly for all points in an infinite universe, BARRING THE INTRODUCTION OF EXTRA DIMENSIONS, because for every two points you move apart, they become closer to other points in an equal and opposite way.

No, that cannot happen uniformly for all points in an infinite universe, BARRING THE INTRODUCTION OF EXTRA DIMENSIONS, because for every two points you move apart, they become closer to other points in an equal and opposite way.

CaptainToonces

2010-May-19, 08:17 AM

No, Pacman space is 2D. It is flat. So it is a simple tiled space.

You used Pacman as an example because of the passages that link left and right, correct? This shape indeed defines a cylinder. It is not 2d.

You used Pacman as an example because of the passages that link left and right, correct? This shape indeed defines a cylinder. It is not 2d.

Strange

2010-May-19, 08:28 AM

Thanks to Jeff Root for the only sensible answer of them all :)

What a lot of you seem to be overlooking, barring perhaps the introduction of extra dimensions, is that the word "expands" itself implies a center, because expands means to gain volume, and anything with volume has a center, so you cannot say that the universe is expanding without saying it has a center. I think the correct answer to my question, is really, "we don't know."

Despite the cryptic final comment, undidly's analogy is a good one: say your existence is confined to the surface of a sphere (i.e. a 2D analogy for a 3D universe). If the sphere is expanding; the surface (which is all you know) is also expanding but it has no center. All points on the surface of the sphere are moving apart from one another. Except for some local motion, none are moving closer together. The surface is finit, expanind and has no center. That may be the way the universe is.

What a lot of you seem to be overlooking, barring perhaps the introduction of extra dimensions, is that the word "expands" itself implies a center, because expands means to gain volume, and anything with volume has a center, so you cannot say that the universe is expanding without saying it has a center. I think the correct answer to my question, is really, "we don't know."

Despite the cryptic final comment, undidly's analogy is a good one: say your existence is confined to the surface of a sphere (i.e. a 2D analogy for a 3D universe). If the sphere is expanding; the surface (which is all you know) is also expanding but it has no center. All points on the surface of the sphere are moving apart from one another. Except for some local motion, none are moving closer together. The surface is finit, expanind and has no center. That may be the way the universe is.

CaptainToonces

2010-May-19, 08:48 AM

That may be the way the universe is.

yes, it may be. But notice i said "barring the introduction of extra dimensions." When you offer the example of a 2d perception of a 3d environment (the 2d people on the surface of the balloon who don't know it's a 3d balloon) you are introducing the concept of the unknown dimension. But even in that example, there IS a center of the universe, even if that center is NOT IN the universe!

yes, it may be. But notice i said "barring the introduction of extra dimensions." When you offer the example of a 2d perception of a 3d environment (the 2d people on the surface of the balloon who don't know it's a 3d balloon) you are introducing the concept of the unknown dimension. But even in that example, there IS a center of the universe, even if that center is NOT IN the universe!

CaptainToonces

2010-May-19, 08:52 AM

I think the correct answer to my question, is really, "we don't know."

Or maybe the answer could be: in the direction of spacetime that points directly into the past

Or maybe the answer could be: in the direction of spacetime that points directly into the past

Strange

2010-May-19, 09:04 AM

yes, it may be. But notice i said "barring the introduction of extra dimensions." When you offer the example of a 2d perception of a 3d environment (the 2d people on the surface of the balloon who don't know it's a 3d balloon) you are introducing the concept of the unknown dimension. But even in that example, there IS a center of the universe, even if that center is NOT IN the universe!

I would be argue that the third dimension in the balloon analogy has no counterpart in the "real world" - this is where the analogy runs out of steam. (All analogies have a limit; that is why they are analogies.) On the other hand, you could consider the inside of the balloon as the past (where all the smaller balloons existed) in which case, the "center of the universe" is in the past; at the big bang.

I would be argue that the third dimension in the balloon analogy has no counterpart in the "real world" - this is where the analogy runs out of steam. (All analogies have a limit; that is why they are analogies.) On the other hand, you could consider the inside of the balloon as the past (where all the smaller balloons existed) in which case, the "center of the universe" is in the past; at the big bang.

astromark

2010-May-19, 09:45 AM

I like my science to be a bit of this, that, and the other. " In what direction would the center of the Universe be ?"

Can not and does not fit my demanding criteria. You can not go look at the beginning of the universe. You can not even show me all of it now.

That more than 70% of it is some sort of unknown 'dark mater'. That the expansion rate is eccelorating by a force we call 'Dark Energy'

It would seem that a awful lot of things are unknown to us...

It makes perfect sense to me that as time did not begin until the universe had begun to expand. That nano second of time when expansion driven by I know not what began. Was also the beginning of time. To be a part of that whole makes it impossible to define a center. There simply could not be one.

I want good science to show me why I might be wrong. I am looking for a argument that has more than ifs and maybe's. On that base we can not answer this question with anything other than we have little idea. Until we know of other possible conclusions we can only best guess from what we do...

Can not and does not fit my demanding criteria. You can not go look at the beginning of the universe. You can not even show me all of it now.

That more than 70% of it is some sort of unknown 'dark mater'. That the expansion rate is eccelorating by a force we call 'Dark Energy'

It would seem that a awful lot of things are unknown to us...

It makes perfect sense to me that as time did not begin until the universe had begun to expand. That nano second of time when expansion driven by I know not what began. Was also the beginning of time. To be a part of that whole makes it impossible to define a center. There simply could not be one.

I want good science to show me why I might be wrong. I am looking for a argument that has more than ifs and maybe's. On that base we can not answer this question with anything other than we have little idea. Until we know of other possible conclusions we can only best guess from what we do...

Aik4on

2010-May-19, 10:44 AM

The problem in these discussions is always that questions like 'in what direction is the centre of the universe' or 'what shape is the universe?' are inappropriate in the same way that to ask 'what shape is water?' or 'how long is air?' would be generally inappropriate. It seems the structure of the universe cannot be measured, described or accurately visualised in terms of a 3D object. Analogies like the balloon analogy are only slightly useful in demonstrating certain properties of the universe, like expansion.

Shaula

2010-May-19, 12:14 PM

You used Pacman as an example because of the passages that link left and right, correct? This shape indeed defines a cylinder. It is not 2d.

I intended you to be able to move off the screen in any direction and still come back in the corresponding place on the other side of the screen. Maybe I should have said Kwik Snax space but that would have been showing my age. Let's not get caught up in this analogy, OK? I intended it as a 2D analogue of what was going on where to the person embedded in the universe it looks like an infinite tiling series. You could warp it to a 3D shape to 'explain' how it is not truly a tiled universe but that would be introducing extra dimensions that are not visible in the universe postulated...

In essence your question had too many assumptions in it. Or assumptions you would have to make to get an answer. Personally I think from the cosmology I have done that most of the ones you are making to get an answer are wrong (that expansion works as you say it does is a critical one - the universe is not expanding into something and just saying that you don't want there to be higher dimensions doesn't make that go away). The answer to your question is that as far as we understand the universe doesn't have a centre.

I intended you to be able to move off the screen in any direction and still come back in the corresponding place on the other side of the screen. Maybe I should have said Kwik Snax space but that would have been showing my age. Let's not get caught up in this analogy, OK? I intended it as a 2D analogue of what was going on where to the person embedded in the universe it looks like an infinite tiling series. You could warp it to a 3D shape to 'explain' how it is not truly a tiled universe but that would be introducing extra dimensions that are not visible in the universe postulated...

In essence your question had too many assumptions in it. Or assumptions you would have to make to get an answer. Personally I think from the cosmology I have done that most of the ones you are making to get an answer are wrong (that expansion works as you say it does is a critical one - the universe is not expanding into something and just saying that you don't want there to be higher dimensions doesn't make that go away). The answer to your question is that as far as we understand the universe doesn't have a centre.

cosmocrazy

2010-May-19, 02:52 PM

The problem in these discussions is always that questions like 'in what direction is the centre of the universe' or 'what shape is the universe?' are inappropriate in the same way that to ask 'what shape is water?' or 'how long is air?' would be generally inappropriate. It seems the structure of the universe cannot be measured, described or accurately visualised in terms of a 3D object. Analogies like the balloon analogy are only slightly useful in demonstrating certain properties of the universe, like expansion.

And this is why we as yet, if at all, struggle to define a centre point for the universe. The mis-conception lies in imagining the universe as a bubble expanding, and with the balloon analogy people tend to interpret our 3D visualisation of the balloon thinking about the inside of the balloon rather than sticking with the 2D observation instead. Maybe to find a co-ordinate for the centre of our 3D universe would require adding an extra dimension. Well all we have at present is "time" so I guess that is all we have to go on, so maybe the "centre" is back in time at the point of creation.

And this is why we as yet, if at all, struggle to define a centre point for the universe. The mis-conception lies in imagining the universe as a bubble expanding, and with the balloon analogy people tend to interpret our 3D visualisation of the balloon thinking about the inside of the balloon rather than sticking with the 2D observation instead. Maybe to find a co-ordinate for the centre of our 3D universe would require adding an extra dimension. Well all we have at present is "time" so I guess that is all we have to go on, so maybe the "centre" is back in time at the point of creation.

starmountainkid

2010-May-19, 04:46 PM

Thought I'd put my 2cents in here as well. In the 2D analogy, you're asking where is the center of the surface of the Earth? Also, The Theory of Relativity states that all points in space are of equal validity.

DrRocket

2010-May-19, 07:30 PM

The Theory of Relativity states that all points in space are of equal validity.

What is that supposed to mean ? I doubt very much that Einstein could make sense of this, let alone agree with it.

What is that supposed to mean ? I doubt very much that Einstein could make sense of this, let alone agree with it.

Jeff Root

2010-May-19, 07:36 PM

What a lot of you seem to be overlooking, barring perhaps the

introduction of extra dimensions, is that the word "expands" itself

implies a center, because expands means to gain volume, and

anything with volume has a center, so you cannot say that the

universe is expanding without saying it has a center.

Weirdly, the Universe could be expanding without a center, and

without additional spatial dimensions. Relativity theory shows

that the dimension of time is interconvertible with the three

dimensions of space, the mixture being called "spacetime".

By exchanging some space for time, 3-D space can be "curved"

without any 4th spatial dimension.

Mathematically, an infinite volume can "expand" in the sense

that all the bits of stuff in the volume can move away from each

other without any moving closer to each other. Look for "Hilbert

Hotel" for an explanation. However, I think this mathematical

possibility is unphysical, and cannot describe the actual Universe.

While there is no observational evidence that the Universe is

finite, I am certain that the Universe, or the portion of it that was

involved in the Big Bang and is still participating in the expansion,

must be finite. That conclusion is based on the requirement that

everything participating in the expansion must be causally

connected to the origin of the expansion, which was a finite time

in the past.

We don't know whether the Universe has a center. If it *does*

have a center, we don't know whether the center is in a particular

direction. If it *is* in a particular direction, we see no indication

of it. But general relativity theory provides an accurate description

of an expanding Universe 13.7 billion years old which has no center.

Since general relativity has been shown to be accurate in every

test that it has been subjected to, and since every observation

of the Universe is consistent with GR's description of the cosmic

expansion, it is the currently-favored description. That is why

many people insist that the Universe has no center.

-- Jeff, in Minneapolis

introduction of extra dimensions, is that the word "expands" itself

implies a center, because expands means to gain volume, and

anything with volume has a center, so you cannot say that the

universe is expanding without saying it has a center.

Weirdly, the Universe could be expanding without a center, and

without additional spatial dimensions. Relativity theory shows

that the dimension of time is interconvertible with the three

dimensions of space, the mixture being called "spacetime".

By exchanging some space for time, 3-D space can be "curved"

without any 4th spatial dimension.

Mathematically, an infinite volume can "expand" in the sense

that all the bits of stuff in the volume can move away from each

other without any moving closer to each other. Look for "Hilbert

Hotel" for an explanation. However, I think this mathematical

possibility is unphysical, and cannot describe the actual Universe.

While there is no observational evidence that the Universe is

finite, I am certain that the Universe, or the portion of it that was

involved in the Big Bang and is still participating in the expansion,

must be finite. That conclusion is based on the requirement that

everything participating in the expansion must be causally

connected to the origin of the expansion, which was a finite time

in the past.

We don't know whether the Universe has a center. If it *does*

have a center, we don't know whether the center is in a particular

direction. If it *is* in a particular direction, we see no indication

of it. But general relativity theory provides an accurate description

of an expanding Universe 13.7 billion years old which has no center.

Since general relativity has been shown to be accurate in every

test that it has been subjected to, and since every observation

of the Universe is consistent with GR's description of the cosmic

expansion, it is the currently-favored description. That is why

many people insist that the Universe has no center.

-- Jeff, in Minneapolis

Jeff Root

2010-May-19, 08:03 PM

My interpretation of the balloon analogy is that the skin of the

balloon represents a two-dimensional plane cut through the

Universe. Other balloons could represent other plane slices.

The balloons could be of any size and in any position, and would

not need to be concentric. Instead, they would interpenetrate.

Alternatively, the skin of the balloon could be a two-dimensional

mapping of all of three-dimensional space. This would be similar

to a photograph of the night sky, with stars that are actually at

all different distances represented together in the same plane.

By doing this, the entire Universe is represented by the balloon's

skin, but with a confusion-causing compression from 3-D to 2-D.

-- Jeff, in Minneapolis

balloon represents a two-dimensional plane cut through the

Universe. Other balloons could represent other plane slices.

The balloons could be of any size and in any position, and would

not need to be concentric. Instead, they would interpenetrate.

Alternatively, the skin of the balloon could be a two-dimensional

mapping of all of three-dimensional space. This would be similar

to a photograph of the night sky, with stars that are actually at

all different distances represented together in the same plane.

By doing this, the entire Universe is represented by the balloon's

skin, but with a confusion-causing compression from 3-D to 2-D.

-- Jeff, in Minneapolis

DrRocket

2010-May-19, 08:16 PM

Thanks to Jeff Root for the only sensible answer of them all :)

What a lot of you seem to be overlooking, barring perhaps the introduction of extra dimensions, is that the word "expands" itself implies a center, because expands means to gain volume, and anything with volume has a center, so you cannot say that the universe is expanding without saying it has a center. I think the correct answer to my question, is really, "we don't know."

No.

Imagine a rubber sheet, infinite in extent. It has no well-defined center. Paint grid lines on it. Now imagine that it is being uniformly stretched. It is expanding and the grid lines are moving apart. There is no center and no center of expansion either.

What a lot of you seem to be overlooking, barring perhaps the introduction of extra dimensions, is that the word "expands" itself implies a center, because expands means to gain volume, and anything with volume has a center, so you cannot say that the universe is expanding without saying it has a center. I think the correct answer to my question, is really, "we don't know."

No.

Imagine a rubber sheet, infinite in extent. It has no well-defined center. Paint grid lines on it. Now imagine that it is being uniformly stretched. It is expanding and the grid lines are moving apart. There is no center and no center of expansion either.

Jeff Root

2010-May-19, 08:19 PM

A Pacman game level is topologically a cylinder. It is not

geometrically cylindrical. Points on a screen are connected

together logically the same way as points on a cylinder are

connected, but the distances and angles do not necessarily

match those on a cylinder.

A physical cylinder has a geometrically cylindrical surface.

The cylinder has a center, but its surface does not. If the

surface of a cylinder is used to represent the Universe, the

center of the cylinder is most likely not intended to have

any significance. Sometimes the only significant feature is

the topology of the surface, but sometimes the geometry

is what is being demonstrated.

-- Jeff, in Minneapolis

geometrically cylindrical. Points on a screen are connected

together logically the same way as points on a cylinder are

connected, but the distances and angles do not necessarily

match those on a cylinder.

A physical cylinder has a geometrically cylindrical surface.

The cylinder has a center, but its surface does not. If the

surface of a cylinder is used to represent the Universe, the

center of the cylinder is most likely not intended to have

any significance. Sometimes the only significant feature is

the topology of the surface, but sometimes the geometry

is what is being demonstrated.

-- Jeff, in Minneapolis

DrRocket

2010-May-19, 08:24 PM

A Pacman game level is topologically a cylinder. It is not

geometrically cylidrical. Points on a screen are connected

together logically the same way as points on a cylinder are

connected, but the distances and angles do not necessarily

match those on a cylinder.

A physical cylinder has a geometrically cylindrical surface.

The cylinder has a center, but its surface does not. If the

surface of a cylinder is used to represent the Universe, the

center of the cylinder is most likely not intended to have

any significance. Sometimes the only significant feature is

the topology of the surface, but sometimes the geometry

is what is being demonstrated.

-- Jeff, in Minneapolis

A PacMan game is a topological torus. Geometrically it is a flat torus.

geometrically cylidrical. Points on a screen are connected

together logically the same way as points on a cylinder are

connected, but the distances and angles do not necessarily

match those on a cylinder.

A physical cylinder has a geometrically cylindrical surface.

The cylinder has a center, but its surface does not. If the

surface of a cylinder is used to represent the Universe, the

center of the cylinder is most likely not intended to have

any significance. Sometimes the only significant feature is

the topology of the surface, but sometimes the geometry

is what is being demonstrated.

-- Jeff, in Minneapolis

A PacMan game is a topological torus. Geometrically it is a flat torus.

astromark

2010-May-19, 08:45 PM

For these reasons I like to think of the expanding yeast rich loaf of cooking raisin bread. Where each and every object suspended in that dough is getting more distant from each other... That the loaf being so large that no edge is detectable... No center can be found..

Understanding that this does not prohibit there being one. Its at least a 3d image... So I put aside the balloons analogy as getting a little complicated when you start overlapping conjoined interacting ...coffee...:oI need coffee....

Understanding that this does not prohibit there being one. Its at least a 3d image... So I put aside the balloons analogy as getting a little complicated when you start overlapping conjoined interacting ...coffee...:oI need coffee....

DrRocket

2010-May-19, 08:46 PM

For these reasons I like to think of the expanding yeast rich loaf of cooking raisin bread. Where each and every object suspended in that dough is getting more distant from each other... That the loaf being so large that no edge is detectable... No center can be found..

Understanding that this does not prohibit there being one. Its at least a 3d image... So I put aside the balloons analogy as getting a little complicated when you start overlapping conjoined interacting ...coffee...:oI need coffee....

Irish coffee ?

Understanding that this does not prohibit there being one. Its at least a 3d image... So I put aside the balloons analogy as getting a little complicated when you start overlapping conjoined interacting ...coffee...:oI need coffee....

Irish coffee ?

Jeff Root

2010-May-19, 08:59 PM

The problem in these discussions is always that questions like

'in what direction is the centre of the universe' or 'what shape is

the universe?' are inappropriate in the same way that to ask

'what shape is water?' or 'how long is air?' would be generally

inappropriate.

I disagree.

The questions are completely appropriate. They are completely

natural and obvious questions. It is the strangeness of the answers

which makes the questions seem inappropriate. You cannot answer

the questions as asked, so you criticise the question. That isn't

necessary, or appropriate.

I believe there are stupid questions, but these aren't.

It seems the structure of the universe cannot be measured, described or

accurately visualised in terms of a 3D object. Analogies like the balloon

analogy are only slightly useful in demonstrating certain properties of the

universe, like expansion.

The balloon analogy can demonstrate:

- How clusters of galaxies move apart

- How individual galaxies don't expand

- How galaxies within a cluster don't expand

- How the expansion rate increases with separation

- How the expansion rate changes with time

- How the expansion can be centerless

- How light can be redshifted by cosmic expansion

- How light travel times can depend on expansion

- The topology of a closed Universe

- The geometry of a Universe with positive curvature

This list is not exhaustive. Some features are harder to show

than others.

-- Jeff, in Minneapolis

'in what direction is the centre of the universe' or 'what shape is

the universe?' are inappropriate in the same way that to ask

'what shape is water?' or 'how long is air?' would be generally

inappropriate.

I disagree.

The questions are completely appropriate. They are completely

natural and obvious questions. It is the strangeness of the answers

which makes the questions seem inappropriate. You cannot answer

the questions as asked, so you criticise the question. That isn't

necessary, or appropriate.

I believe there are stupid questions, but these aren't.

It seems the structure of the universe cannot be measured, described or

accurately visualised in terms of a 3D object. Analogies like the balloon

analogy are only slightly useful in demonstrating certain properties of the

universe, like expansion.

The balloon analogy can demonstrate:

- How clusters of galaxies move apart

- How individual galaxies don't expand

- How galaxies within a cluster don't expand

- How the expansion rate increases with separation

- How the expansion rate changes with time

- How the expansion can be centerless

- How light can be redshifted by cosmic expansion

- How light travel times can depend on expansion

- The topology of a closed Universe

- The geometry of a Universe with positive curvature

This list is not exhaustive. Some features are harder to show

than others.

-- Jeff, in Minneapolis

CaptainToonces

2010-May-20, 05:47 PM

Imagine a rubber sheet, infinite in extent. Now imagine that it is being uniformly stretched.

You're not grasping just how significant adding infinity to the equation is. To say "stretch an infinite 2D sheet" is nonsensical. You go to attach your stretching clamps to the edges, and "hmm now where it is that edge"? I know you think you can imagine grid lines "expanding" but you need to understand that your imagination is applying a non-infinity concept there and it is only by this that you can visualize this expansion to occur.

I personally think that the universe will never be proven to be finite, and it is definitely very difficult to say the line in spacetime extending directly into the future is finite.

You're not grasping just how significant adding infinity to the equation is. To say "stretch an infinite 2D sheet" is nonsensical. You go to attach your stretching clamps to the edges, and "hmm now where it is that edge"? I know you think you can imagine grid lines "expanding" but you need to understand that your imagination is applying a non-infinity concept there and it is only by this that you can visualize this expansion to occur.

I personally think that the universe will never be proven to be finite, and it is definitely very difficult to say the line in spacetime extending directly into the future is finite.

Hornblower

2010-May-20, 06:45 PM

You're not grasping just how significant adding infinity to the equation is. To say "stretch an infinite 2D sheet" is nonsensical. You go to attach your stretching clamps to the edges, and "hmm now where it is that edge"? I know you think you can imagine grid lines "expanding" but you need to understand that your imagination is applying a non-infinity concept there and it is only by this that you can visualize this expansion to occur.

I personally think that the universe will never be proven to be finite, and it is definitely very difficult to say the line in spacetime extending directly into the future is finite.

Please explain what you mean, in appropriate rigorous mathematical detail, by "non-infinity concept", and how it applies to this line of thought.

I personally think that the universe will never be proven to be finite, and it is definitely very difficult to say the line in spacetime extending directly into the future is finite.

Please explain what you mean, in appropriate rigorous mathematical detail, by "non-infinity concept", and how it applies to this line of thought.

astromark

2010-May-20, 08:02 PM

" In what direction is the center of the Universe ?" I could say its over there.... but I do not know which direction to point.

That little figure 8 that looks like its fallen over which represents infinity. Interfering with the conceptual image... It 'is' hard to imagine. Try harder.

Try to picture the most distant objects we have knowledge of. Those very faint smudges on the Hubble Deep Field. Soon to be unseen as the space between them and us is expanding ever faster. That light image we have built will become fainter as that distance increases.

We have no chance of ever seeing more than is apparent today. Not a particularly accurate but, in understanding the cosmoligy of a eccelorating expansion its the truth.

The only answer I can be sure of is this...' Dependant upon the knowledge we have. As to a closed finite space expanding. There could be calculated a central point of all mass.' or Not. Entirely dependent upon your ability to perceive a concept.

If how ever that we could show the universe to be unbound and continuing in that expansion and rate for infinity... then No.

Umm.. Err.. Oh yaa.. thats perfectly clear....

and no, my coffee is fresh and pure, plunger type, No sugar or milk.

That little figure 8 that looks like its fallen over which represents infinity. Interfering with the conceptual image... It 'is' hard to imagine. Try harder.

Try to picture the most distant objects we have knowledge of. Those very faint smudges on the Hubble Deep Field. Soon to be unseen as the space between them and us is expanding ever faster. That light image we have built will become fainter as that distance increases.

We have no chance of ever seeing more than is apparent today. Not a particularly accurate but, in understanding the cosmoligy of a eccelorating expansion its the truth.

The only answer I can be sure of is this...' Dependant upon the knowledge we have. As to a closed finite space expanding. There could be calculated a central point of all mass.' or Not. Entirely dependent upon your ability to perceive a concept.

If how ever that we could show the universe to be unbound and continuing in that expansion and rate for infinity... then No.

Umm.. Err.. Oh yaa.. thats perfectly clear....

and no, my coffee is fresh and pure, plunger type, No sugar or milk.

Aik4on

2010-May-20, 08:11 PM

I disagree.

The questions are completely appropriate. They are completely

natural and obvious questions. It is the strangeness of the answers

which makes the questions seem inappropriate.

Yes totally natural just as it is completely natural for us to have the innate perception of a spatially 3D universe and, from this extrapolate a visualisation of the universe, globally, as a 3D object. Perhaps the question could be more fairly described as inadequate than inappropriate.

As for the benefits you list for the balloon model, I don't disagree but I do think that this model has promulgated a widely held misconception of a 'bubble shaped' universe expanding into 'empty space' or matter expanding into space from a point. I much prefer astromarks raisin loaf analogy.

The questions are completely appropriate. They are completely

natural and obvious questions. It is the strangeness of the answers

which makes the questions seem inappropriate.

Yes totally natural just as it is completely natural for us to have the innate perception of a spatially 3D universe and, from this extrapolate a visualisation of the universe, globally, as a 3D object. Perhaps the question could be more fairly described as inadequate than inappropriate.

As for the benefits you list for the balloon model, I don't disagree but I do think that this model has promulgated a widely held misconception of a 'bubble shaped' universe expanding into 'empty space' or matter expanding into space from a point. I much prefer astromarks raisin loaf analogy.

DrRocket

2010-May-20, 08:17 PM

You're not grasping just how significant adding infinity to the equation is. To say "stretch an infinite 2D sheet" is nonsensical. You go to attach your stretching clamps to the edges, and "hmm now where it is that edge"? I know you think you can imagine grid lines "expanding" but you need to understand that your imagination is applying a non-infinity concept there and it is only by this that you can visualize this expansion to occur.

I personally think that the universe will never be proven to be finite, and it is definitely very difficult to say the line in spacetime extending directly into the future is finite.

wrong

I am rather well versed in dealing with infinity. This is pretty standard stuff. You need to learn some mathematics.

I personally think that the universe will never be proven to be finite, and it is definitely very difficult to say the line in spacetime extending directly into the future is finite.

wrong

I am rather well versed in dealing with infinity. This is pretty standard stuff. You need to learn some mathematics.

DrRocket

2010-May-20, 08:20 PM

Yes totally natural just as it is completely natural for us to have the innate perception of a spatially 3D universe and, from this extrapolate a visualisation of the universe, globally, as a 3D object. Perhaps the question could be more fairly described as inadequate than inappropriate.

As for the benefits you list for the balloon model, I don't disagree but I do think that this model has promulgated a widely held misconception of a 'bubble shaped' universe expanding into 'empty space' or matter expanding into space from a point. I much prefer astromarks raisin loaf analogy.

The purpose of the balloon analogy is to try to convey the notion of a manifold without actually using the relevant mathematics. If you view it that way it works pretty well (as a 2-dimensional analogy). But if you somehow look at at it as sphere embedded in 3-space the analogy is lost, and the imagery fails miserably.

The universe did not expand into empty space. It is space that is expanding, and it is not "in" anything. That is the point of an intrinsic manifold as opposed to an embedded manifold.

As for the benefits you list for the balloon model, I don't disagree but I do think that this model has promulgated a widely held misconception of a 'bubble shaped' universe expanding into 'empty space' or matter expanding into space from a point. I much prefer astromarks raisin loaf analogy.

The purpose of the balloon analogy is to try to convey the notion of a manifold without actually using the relevant mathematics. If you view it that way it works pretty well (as a 2-dimensional analogy). But if you somehow look at at it as sphere embedded in 3-space the analogy is lost, and the imagery fails miserably.

The universe did not expand into empty space. It is space that is expanding, and it is not "in" anything. That is the point of an intrinsic manifold as opposed to an embedded manifold.

mugaliens

2010-May-20, 08:58 PM

Yes totally natural just as it is completely natural for us to have the innate perception of a spatially 3D universe and, from this extrapolate a visualisation of the universe, globally, as a 3D object.

Yes, except for one thing: The balloon analogy isn't a 3D object. It's a curved 2D surface. A more appropriate analogy would be a spherical sponge which expands when wetted.

As for the benefits you list for the balloon model, I don't disagree but I do think that this model has promulgated a widely held misconception of a 'bubble shaped' universe expanding into 'empty space' or matter expanding into space from a point. I much prefer astromarks raisin loaf analogy.

Yeah, I'd go with a raisin loaf - much more tasty than a sponge!

Yes, except for one thing: The balloon analogy isn't a 3D object. It's a curved 2D surface. A more appropriate analogy would be a spherical sponge which expands when wetted.

As for the benefits you list for the balloon model, I don't disagree but I do think that this model has promulgated a widely held misconception of a 'bubble shaped' universe expanding into 'empty space' or matter expanding into space from a point. I much prefer astromarks raisin loaf analogy.

Yeah, I'd go with a raisin loaf - much more tasty than a sponge!

astromark

2010-May-20, 09:39 PM

So it is just a mater of perception. Can you visualize a truly infinite expanding ever faster Universe. Because if you can not and insist on a hanging in space sort of analogy. Understanding that the Universe is not in space. It is space with very little stuff in it... Then you will never see the big picture we are attempting to explain.

Now I must buy fresh hot raisin bread and coffee....

Now I must buy fresh hot raisin bread and coffee....

Jeff Root

2010-May-20, 10:14 PM

Imagine a rubber sheet, infinite in extent. Now imagine that it is being

uniformly stretched.

You're not grasping just how significant adding infinity to the equation is.

I'm replying to this before having read any of the posts which follow it.

I presume that DrRocket has already replied.

You're not grasping that "DrRocket" is an actual, bona fide mathematician.

The problem you raise is elementary to him. You can find explanations of

how something infinite can expand by looking up "Hilbert hotel".

-- Jeff, in Minneapolis

uniformly stretched.

You're not grasping just how significant adding infinity to the equation is.

I'm replying to this before having read any of the posts which follow it.

I presume that DrRocket has already replied.

You're not grasping that "DrRocket" is an actual, bona fide mathematician.

The problem you raise is elementary to him. You can find explanations of

how something infinite can expand by looking up "Hilbert hotel".

-- Jeff, in Minneapolis

Jeff Root

2010-May-20, 10:39 PM

The universe did not expand into empty space. It is space that

is expanding, and it is not "in" anything.

I understand that that is how general relativity describes the

cosmic expansion, and I agree that that description fits the

observations. But I would like to see a fairly complete outline

of the arguments for the assertions in those two sentences.

I have read books about cosmology, yet I have never seen an

argument supporting those assertions. I am skeptical of their

truth. My view is that they are possibilities, not known facts.

-- Jeff, in Minneapolis

is expanding, and it is not "in" anything.

I understand that that is how general relativity describes the

cosmic expansion, and I agree that that description fits the

observations. But I would like to see a fairly complete outline

of the arguments for the assertions in those two sentences.

I have read books about cosmology, yet I have never seen an

argument supporting those assertions. I am skeptical of their

truth. My view is that they are possibilities, not known facts.

-- Jeff, in Minneapolis

DrRocket

2010-May-20, 11:04 PM

I understand that that is how general relativity describes the

cosmic expansion, and I agree that that description fits the

observations. But I would like to see a fairly complete outline

of the arguments for the assertions in those two sentences.

I have read books about cosmology, yet I have never seen an

argument supporting those assertions. I am skeptical of their

truth. My view is that they are possibilities, not known facts.

-- Jeff, in Minneapolis

General relativity models spacetime as a 4-manifold with a Lorentzian metric. That is the starting point for the theory. After that, one has the Einstein field equations that determine the curvature tensor of the manifold as function of the distribution of mass-energy.

But the basic structure is a 4-manifold and that manifold is not embedded in anything. That manifold includes all of space and all of time, all mixed together via the curvature.

The question is then what does it mean to say that "space is expanding". To do this one realizes the entire spacetime manifold as a foliation of 3-dimensional "slices" parameterized by a single parameter, which is "time". This can be done under certain assumptions, homogeneity and isotropy. That gives one a set of non-linear "coordinates", with one coordinate being time. In these coordinates, points with the same spatial coordinates move apart as the time parameter increases, and that is what is meant by "space is expanding". This does not require space to be "in" anything" or to be expanding into anything.

A simple analogy would be to imagine a cone. The axis of the cone is time, and circles normal to the axis represent space (a 1-dimensional analogy). So as you go "up" the time axis the diameter of the circles increases -- space is expanding. But the circles are still on the cone, which is in this analogy all of spacetime.

The other way to look at it would be to simply note that universe is, by definition, the whole enchilada. If it were in something larger, and if that something larger could influence the universe, then it would be the universe. In short, if there was anywhere for the universe to go that somewhere else would actually be part of the universe.

Now, if you wish, there is a mathematical theorem that states that any Lorentzian manifold can be embedded in a Euclidean space of some sufficiently large dimension with metric of appropriate signature, but that is a purely mathematical result, and adds nothing to the general theory of relativity. In fact this perspective is detrimental to the philosophy of "general covariance" that was the impetus for Einstein's formulation of general relativity. So, it is an unnecessary concept and it actually obscures what is going on.

Virtually nothing in cosmology is a known fact. A critical element to all of cosmology is the cosmological principle -- the assumption that the universe on the largest scale is homogeneous and isotropic. All observational evidence is consistent with this assumption, but it is an assumption and given the size of the universe it is not provable. Modern cosmology is dependent on general relativity, so if one assumes that there is some problem with general relativity, that problem automatically applies to cosmology.

On the other hand the very notion of "expansion of space" is based on general relativity, so if one rejects general relativity as the context for understanding the expansion then there a lot more important issues than whether or not it is expanding into something.

cosmic expansion, and I agree that that description fits the

observations. But I would like to see a fairly complete outline

of the arguments for the assertions in those two sentences.

I have read books about cosmology, yet I have never seen an

argument supporting those assertions. I am skeptical of their

truth. My view is that they are possibilities, not known facts.

-- Jeff, in Minneapolis

General relativity models spacetime as a 4-manifold with a Lorentzian metric. That is the starting point for the theory. After that, one has the Einstein field equations that determine the curvature tensor of the manifold as function of the distribution of mass-energy.

But the basic structure is a 4-manifold and that manifold is not embedded in anything. That manifold includes all of space and all of time, all mixed together via the curvature.

The question is then what does it mean to say that "space is expanding". To do this one realizes the entire spacetime manifold as a foliation of 3-dimensional "slices" parameterized by a single parameter, which is "time". This can be done under certain assumptions, homogeneity and isotropy. That gives one a set of non-linear "coordinates", with one coordinate being time. In these coordinates, points with the same spatial coordinates move apart as the time parameter increases, and that is what is meant by "space is expanding". This does not require space to be "in" anything" or to be expanding into anything.

A simple analogy would be to imagine a cone. The axis of the cone is time, and circles normal to the axis represent space (a 1-dimensional analogy). So as you go "up" the time axis the diameter of the circles increases -- space is expanding. But the circles are still on the cone, which is in this analogy all of spacetime.

The other way to look at it would be to simply note that universe is, by definition, the whole enchilada. If it were in something larger, and if that something larger could influence the universe, then it would be the universe. In short, if there was anywhere for the universe to go that somewhere else would actually be part of the universe.

Now, if you wish, there is a mathematical theorem that states that any Lorentzian manifold can be embedded in a Euclidean space of some sufficiently large dimension with metric of appropriate signature, but that is a purely mathematical result, and adds nothing to the general theory of relativity. In fact this perspective is detrimental to the philosophy of "general covariance" that was the impetus for Einstein's formulation of general relativity. So, it is an unnecessary concept and it actually obscures what is going on.

Virtually nothing in cosmology is a known fact. A critical element to all of cosmology is the cosmological principle -- the assumption that the universe on the largest scale is homogeneous and isotropic. All observational evidence is consistent with this assumption, but it is an assumption and given the size of the universe it is not provable. Modern cosmology is dependent on general relativity, so if one assumes that there is some problem with general relativity, that problem automatically applies to cosmology.

On the other hand the very notion of "expansion of space" is based on general relativity, so if one rejects general relativity as the context for understanding the expansion then there a lot more important issues than whether or not it is expanding into something.

DrRocket

2010-May-20, 11:16 PM

It occurs to me that some of the problems with gaining an understanding of the expansion of space are due to intuition based on ordinary Newtonian mechanics and everyday experience.

Newtonian mechanics, with which most people are familiar and for which most people have intuition is basically a description of how bodies move through space as a function of time. Space and time are treated as concepts that are intuitively clear and that correspond to everyday experience with clocks and rigid rulers. The theory takes those concepts as rather obvious and then develops a theory of dynamics based on forces and ordinary differential equations. But the main point is that space and time match our intuitive everyday notions.

The special and general theories of Einstein are really different. They do provide a new formulation of dynamics, and some non-intuitive effects like time dilation and length contraction. But fundamentally the theory of relativity is less about dynamics than it is a revolutionary theory of the nature of space and time. So, to even speak of notions like "space is expanding" one is forced to use the models of relativity. Expansion of space is nonsensical in the Newtonian context. For Newton space was fixed an immutable. It is only in Einstein's theory that the statement "space is expanding" has any meaning, let alone is possible.

So, if one want to talk about expansion of space outside of general relativity, then one is obligated to lay the groundwork and explain what "expansion of space" means in that new context. In general relativity, expansion of space is described using the methods of Riemannian geometry on an intrinsic manifold and has nothing to do with expansion into anything outside of the universe. If one wishes to discuss it in some different context, then one must define what that context is.

Newtonian mechanics, with which most people are familiar and for which most people have intuition is basically a description of how bodies move through space as a function of time. Space and time are treated as concepts that are intuitively clear and that correspond to everyday experience with clocks and rigid rulers. The theory takes those concepts as rather obvious and then develops a theory of dynamics based on forces and ordinary differential equations. But the main point is that space and time match our intuitive everyday notions.

The special and general theories of Einstein are really different. They do provide a new formulation of dynamics, and some non-intuitive effects like time dilation and length contraction. But fundamentally the theory of relativity is less about dynamics than it is a revolutionary theory of the nature of space and time. So, to even speak of notions like "space is expanding" one is forced to use the models of relativity. Expansion of space is nonsensical in the Newtonian context. For Newton space was fixed an immutable. It is only in Einstein's theory that the statement "space is expanding" has any meaning, let alone is possible.

So, if one want to talk about expansion of space outside of general relativity, then one is obligated to lay the groundwork and explain what "expansion of space" means in that new context. In general relativity, expansion of space is described using the methods of Riemannian geometry on an intrinsic manifold and has nothing to do with expansion into anything outside of the universe. If one wishes to discuss it in some different context, then one must define what that context is.

CaptainToonces

2010-May-21, 03:17 AM

Please explain what you mean, in appropriate rigorous mathematical detail, by "non-infinity concept", and how it applies to this line of thought.

By "non-infinity concept" I mean any expression that does not contain the notion of infinity. Another way you could say it would be a formula that doesn't include the infinity symbol or a divide by 0.

I figured that would be self-explanatory from the name of the term but I guess not.

I suppose that haggling about the nature of infinity is rather pointless since it is an imaginary concept.

In my understanding of the meanings of infinite and expand, it would seem that if you described something, say a number line, with the word "infinite" you would be implying that this thing is already expanded to the absolute maximum size that it can be expanded to and therefore cannot expand any more.

The Hilbert paradox proves only the nonsensical nature of trying to describe real things using the concept of infinity. You can move guest 1 to room 2, guest 2 to room 3, and so on infinitely, but that is an unclosed loop in your algorithm and if you did this in reality you would never get done ushering these poor guests out of their rooms. If you try to close the loop in the algorithm, you are left with a guest out of a room somewhere down the line.

By "non-infinity concept" I mean any expression that does not contain the notion of infinity. Another way you could say it would be a formula that doesn't include the infinity symbol or a divide by 0.

I figured that would be self-explanatory from the name of the term but I guess not.

I suppose that haggling about the nature of infinity is rather pointless since it is an imaginary concept.

In my understanding of the meanings of infinite and expand, it would seem that if you described something, say a number line, with the word "infinite" you would be implying that this thing is already expanded to the absolute maximum size that it can be expanded to and therefore cannot expand any more.

The Hilbert paradox proves only the nonsensical nature of trying to describe real things using the concept of infinity. You can move guest 1 to room 2, guest 2 to room 3, and so on infinitely, but that is an unclosed loop in your algorithm and if you did this in reality you would never get done ushering these poor guests out of their rooms. If you try to close the loop in the algorithm, you are left with a guest out of a room somewhere down the line.

CaptainToonces

2010-May-21, 03:31 AM

wrong

I am rather well versed in dealing with infinity. This is pretty standard stuff. You need to learn some mathematics.

Oh you are well-versed in dealing with infinity? Impressive. I am an expert on the value of the square root of -1.

Oh please sir, would you teach me the mathematics I need to learn then? What mathematics are you referring to that is standard stuff? I would much rather you describe these mathematics than simply attempting to belittle my understanding of them.

I am rather well versed in dealing with infinity. This is pretty standard stuff. You need to learn some mathematics.

Oh you are well-versed in dealing with infinity? Impressive. I am an expert on the value of the square root of -1.

Oh please sir, would you teach me the mathematics I need to learn then? What mathematics are you referring to that is standard stuff? I would much rather you describe these mathematics than simply attempting to belittle my understanding of them.

Geo Kaplan

2010-May-21, 03:32 AM

The Hilbert paradox proves only the nonsensical nature of trying to describe real things using the concept of infinity. You can move guest 1 to room 2, guest 2 to room 3, and so on infinitely, but that is an unclosed loop in your algorithm and if you did this in reality you would never get done ushering these poor guests out of their rooms. If you try to close the loop in the algorithm, you are left with a guest out of a room somewhere down the line.

Not "nonsensical." Counterintuitive, maybe, but not nonsensical.

"In reality" there are no guests left without a room. That it takes an infinite time to accommodate everyone is besides the point.

Not "nonsensical." Counterintuitive, maybe, but not nonsensical.

"In reality" there are no guests left without a room. That it takes an infinite time to accommodate everyone is besides the point.

CaptainToonces

2010-May-21, 03:37 AM

Not "nonsensical." Counterintuitive, maybe, but not nonsensical.

infinity = nonsensical. They are basically synonyms. So the notion became nonsensical as soon as you said "the hotel has an infinite number of rooms"

I suppose by that same token it is nonsensical for me to say "you can't stretch something that's infinite." So I redact that statement. It's mostly useless to speak of things being infinite.

infinity = nonsensical. They are basically synonyms. So the notion became nonsensical as soon as you said "the hotel has an infinite number of rooms"

I suppose by that same token it is nonsensical for me to say "you can't stretch something that's infinite." So I redact that statement. It's mostly useless to speak of things being infinite.

Jens

2010-May-21, 03:38 AM

You're not grasping just how significant adding infinity to the equation is. To say "stretch an infinite 2D sheet" is nonsensical.

I don't think it is. As an example, suppose we have the set of integers. We can all agree it is infinite. Then you add 1.5 and 2.5 and 3.5. You've effectively doubled the length of the line, but it's still infinite, and expanding.

I don't think it is. As an example, suppose we have the set of integers. We can all agree it is infinite. Then you add 1.5 and 2.5 and 3.5. You've effectively doubled the length of the line, but it's still infinite, and expanding.

DrRocket

2010-May-21, 03:47 AM

By "non-infinity concept" I mean any expression that does not contain the notion of infinity. Another way you could say it would be a formula that doesn't include the infinity symbol or a divide by 0.

I figured that would be self-explanatory from the name of the term but I guess not.

I suppose that haggling about the nature of infinity is rather pointless since it is an imaginary concept.

In my understanding of the meanings of infinite and expand, it would seem that if you described something, say a number line, with the word "infinite" you would be implying that this thing is already expanded to the absolute maximum size that it can be expanded to and therefore cannot expand any more.

The Hilbert paradox proves only the nonsensical nature of trying to describe real things using the concept of infinity. You can move guest 1 to room 2, guest 2 to room 3, and so on infinitely, but that is an unclosed loop in your algorithm and if you did this in reality you would never get done ushering these poor guests out of their rooms. If you try to close the loop in the algorithm, you are left with a guest out of a room somewhere down the line.

Congratulations. You have managed to misunderstand literally everything.

I figured that would be self-explanatory from the name of the term but I guess not.

I suppose that haggling about the nature of infinity is rather pointless since it is an imaginary concept.

In my understanding of the meanings of infinite and expand, it would seem that if you described something, say a number line, with the word "infinite" you would be implying that this thing is already expanded to the absolute maximum size that it can be expanded to and therefore cannot expand any more.

The Hilbert paradox proves only the nonsensical nature of trying to describe real things using the concept of infinity. You can move guest 1 to room 2, guest 2 to room 3, and so on infinitely, but that is an unclosed loop in your algorithm and if you did this in reality you would never get done ushering these poor guests out of their rooms. If you try to close the loop in the algorithm, you are left with a guest out of a room somewhere down the line.

Congratulations. You have managed to misunderstand literally everything.

DrRocket

2010-May-21, 03:49 AM

infinity = nonsensical. They are basically synonyms. So the notion became nonsensical as soon as you said "the hotel has an infinite number of rooms"

I suppose by that same token it is nonsensical for me to say "you can't stretch something that's infinite." So I redact that statement. It's mostly useless to speak of things being infinite.

The fact that you don't understand a concept is not evidence that it is nonsensical. It is only evidence that you do not understand.

You really do need to learn some mathematics.

Take the ordinary Euclidean plane, a plane of vector points. Multiply each point by 2. You have just stretched it. You can create the same effect optically with a magnifying glass.

I suppose by that same token it is nonsensical for me to say "you can't stretch something that's infinite." So I redact that statement. It's mostly useless to speak of things being infinite.

The fact that you don't understand a concept is not evidence that it is nonsensical. It is only evidence that you do not understand.

You really do need to learn some mathematics.

Take the ordinary Euclidean plane, a plane of vector points. Multiply each point by 2. You have just stretched it. You can create the same effect optically with a magnifying glass.

CaptainToonces

2010-May-21, 03:52 AM

It occurs to me that some of the problems with gaining an understanding of the expansion of space are due to intuition based on ordinary Newtonian mechanics and everyday experience.

Not really. I don't think anyone here is arguing for absolute space and time.

The original question of the post was which direction does the center of the universe lie in, which seems like it might be a knowable datum, since the Earth has a center, the Solar System has one, the Galaxy has one, and the fact that we think the universe was once a very small point.

We seem to have gotten a little sidetracked on some semantics about infinity. In fact i haven't seen anyone argue against the suggestion that the direction of the center of the universe is the spacetime 4-vector pointing directly backwards in time (0,0,0,-1).

Not really. I don't think anyone here is arguing for absolute space and time.

The original question of the post was which direction does the center of the universe lie in, which seems like it might be a knowable datum, since the Earth has a center, the Solar System has one, the Galaxy has one, and the fact that we think the universe was once a very small point.

We seem to have gotten a little sidetracked on some semantics about infinity. In fact i haven't seen anyone argue against the suggestion that the direction of the center of the universe is the spacetime 4-vector pointing directly backwards in time (0,0,0,-1).

Geo Kaplan

2010-May-21, 04:00 AM

infinity = nonsensical. They are basically synonyms. So the notion became nonsensical as soon as you said "the hotel has an infinite number of rooms"

These may be synonyms to you, but the universe of ideas if fortunately not limited to what's inside your own cranium. If you were to acquaint yourself with Hilbert's proof, the "paradox" would be less mysterious to you. Georg Cantor's ghost is willing to help you out, too, if you approach the subject with an open mind.

These may be synonyms to you, but the universe of ideas if fortunately not limited to what's inside your own cranium. If you were to acquaint yourself with Hilbert's proof, the "paradox" would be less mysterious to you. Georg Cantor's ghost is willing to help you out, too, if you approach the subject with an open mind.

CaptainToonces

2010-May-21, 04:09 AM

If you were to acquaint yourself with Hilbert's proof, the "paradox" would be less mysterious to you. Georg Cantor's ghost is willing to help you out, too, if you approach the subject with an open mind.

I'm right here, sir, and ready to learn. Enlighten me! How is the idea of moving an infinite number of people from room to room not an infinite loop? Got a link? There's nothing closed-minded here. I stand ready to be proven wrong!

I could do without the rudeness, though, from DrRocket!

I'm right here, sir, and ready to learn. Enlighten me! How is the idea of moving an infinite number of people from room to room not an infinite loop? Got a link? There's nothing closed-minded here. I stand ready to be proven wrong!

I could do without the rudeness, though, from DrRocket!

DrRocket

2010-May-21, 04:09 AM

In fact i haven't seen anyone argue against the suggestion that the direction of the center of the universe is the spacetime 4-vector pointing directly backwards in time (0,0,0,-1).

That is ridiculous. It doesn't even mean anything.

If you think you understand what you are saying then you really need to do a little homework.

That is ridiculous. It doesn't even mean anything.

If you think you understand what you are saying then you really need to do a little homework.

DrRocket

2010-May-21, 04:10 AM

I'm right here, sir, and ready to learn. Enlighten me! How is the idea of moving and infinite number of people from room to room not an infinite loop? Got a link? There's nothing closed-minded here. I stand ready to be proven wrong!

I could do without the rudeness, though, from DrRocket!

That's not right. It's not even wrong -- Wolfgang Pauli

I could do without the rudeness, though, from DrRocket!

That's not right. It's not even wrong -- Wolfgang Pauli

CaptainToonces

2010-May-21, 04:12 AM

I don't think it is. As an example, suppose we have the set of integers. We can all agree it is infinite. Then you add 1.5 and 2.5 and 3.5. You've effectively doubled the length of the line, but it's still infinite, and expanding.

Yeah and if I pencil in millimeter marks to my yardstick it gets longer, right?

Yeah and if I pencil in millimeter marks to my yardstick it gets longer, right?

astromark

2010-May-21, 04:12 AM

I fail to see any argument here... surly expansion is explained by understanding that where objects of mass that are not gravity bound are receding away at a ever increasing rate from one another... expansion. Not of the objects but, of the space. Where is that hard to grasp.

Thats not even the question is it ?

What direction ? All of them. Postulating that one needs to understand higher mathematics is not at issue. A understanding of cooking bread would seem to be all that is required. With a healthy respect for a active imagination. Find the tolerance of understanding and it all becomes clear.

Thats not even the question is it ?

What direction ? All of them. Postulating that one needs to understand higher mathematics is not at issue. A understanding of cooking bread would seem to be all that is required. With a healthy respect for a active imagination. Find the tolerance of understanding and it all becomes clear.

CaptainToonces

2010-May-21, 04:23 AM

In fact i haven't seen anyone argue against the suggestion that the direction of the center of the universe is the spacetime 4-vector pointing directly backwards in time (0,0,0,-1).That is ridiculous. It doesn't even mean anything.

It most definitely means something my good sir! Which part did you not understand? Do you not know what a 4-vector is? Here is the wikipedia article: http://en.wikipedia.org/wiki/4-vector

It most definitely means something my good sir! Which part did you not understand? Do you not know what a 4-vector is? Here is the wikipedia article: http://en.wikipedia.org/wiki/4-vector

Geo Kaplan

2010-May-21, 04:44 AM

I'm right here, sir, and ready to learn. Enlighten me! How is the idea of moving an infinite number of people from room to room not an infinite loop? Got a link? There's nothing closed-minded here. I stand ready to be proven wrong!

Don't move goalposts -- please read what I wrote with the same care I put into writing it. Note, for example, that I explicitly acknowledged that it would take infinite time. I said that there is nothing about this per se that makes it nonsense. It just makes it infinite. There's a difference. Again, just because it is nonsensical to you personally does not make it universally nonsensical. You have overlooked the possibility that the problem lies not with the concept, but with your own understanding.

I really do recommend that you study what Hilbert said about his "paradox", especially given that you cited it specifically. If your mind is truly open, you would also benefit from reading up on what Cantor achieved. Like you, his contemporaries had great difficulties placing concepts of infinity on a rational basis. Cantor showed that this could be done. It takes getting used to, I grant you that, but don't just reject the concept outright because it doesn't make immediate sense to you personally.

ETA: You seem to be laboring under the misapprehension that a program that doesn't execute in finite time is somehow indicative of an underlying irrationality. That is itself an ATM claim, so if you wish to insist on it, I look forward to seeing you start a thread on this in the ATM forum.

Don't move goalposts -- please read what I wrote with the same care I put into writing it. Note, for example, that I explicitly acknowledged that it would take infinite time. I said that there is nothing about this per se that makes it nonsense. It just makes it infinite. There's a difference. Again, just because it is nonsensical to you personally does not make it universally nonsensical. You have overlooked the possibility that the problem lies not with the concept, but with your own understanding.

I really do recommend that you study what Hilbert said about his "paradox", especially given that you cited it specifically. If your mind is truly open, you would also benefit from reading up on what Cantor achieved. Like you, his contemporaries had great difficulties placing concepts of infinity on a rational basis. Cantor showed that this could be done. It takes getting used to, I grant you that, but don't just reject the concept outright because it doesn't make immediate sense to you personally.

ETA: You seem to be laboring under the misapprehension that a program that doesn't execute in finite time is somehow indicative of an underlying irrationality. That is itself an ATM claim, so if you wish to insist on it, I look forward to seeing you start a thread on this in the ATM forum.

astromark

2010-May-21, 04:58 AM

In a other unrelated thread to this one the same Dr Rocket told me with some authority that... 'Time is part of space. Just as space is time related' and that I except without argument. Relativity has a time component. So yes, the center is back there at the beginning...BUT NO its not. Because that does not help you define any direction to a center of all that is... It still is the whole thing regardless of the expansion or timing of when you look.

I see what Dr Rocket is telling you, and agree. Its not about being wrong...and I know about that:o... Its just the language used is not always understood.

I see what Dr Rocket is telling you, and agree. Its not about being wrong...and I know about that:o... Its just the language used is not always understood.

Jens

2010-May-21, 05:00 AM

Yeah and if I pencil in millimeter marks to my yardstick it gets longer, right?

Not if you pencil them in, but if you insert millimeters in, sure it will get longer.

Not if you pencil them in, but if you insert millimeters in, sure it will get longer.

CaptainToonces

2010-May-21, 05:28 AM

Don't move goalposts -- please read what I wrote with the same care I put into writing it. Note, for example, that I explicitly acknowledged that it would take infinite time. I said that there is nothing about this per se that makes it nonsense. It just makes it infinite. There's a difference. Again, just because it is nonsensical to you personally does not make it universally nonsensical. You have overlooked the possibility that the problem lies not with the concept, but with your own understanding.

I really do recommend that you study what Hilbert said about his "paradox", especially given that you cited it specifically. If your mind is truly open, you would also benefit from reading up on what Cantor achieved. Like you, his contemporaries had great difficulties placing concepts of infinity on a rational basis. Cantor showed that this could be done. It takes getting used to, I grant you that, but don't just reject the concept outright because it doesn't make immediate sense to you personally.

ETA: You seem to be laboring under the misapprehension that a program that doesn't execute in finite time is somehow indicative of an underlying irrationality. That is itself an ATM claim, so if you wish to insist on it, I look forward to seeing you start a thread on this in the ATM forum.

Thanks for the recommendation, I'll look into Cantor's diagonal proof of uncountable sets a bit.

Ok you did explicitly mention it would take infinite time. Perhaps something can be rational and infinite at the same time but again i think we are mincing words a bit. Certainly saying that a solution takes infinite time to resolve is somewhat of a drawback!

I really do recommend that you study what Hilbert said about his "paradox", especially given that you cited it specifically. If your mind is truly open, you would also benefit from reading up on what Cantor achieved. Like you, his contemporaries had great difficulties placing concepts of infinity on a rational basis. Cantor showed that this could be done. It takes getting used to, I grant you that, but don't just reject the concept outright because it doesn't make immediate sense to you personally.

ETA: You seem to be laboring under the misapprehension that a program that doesn't execute in finite time is somehow indicative of an underlying irrationality. That is itself an ATM claim, so if you wish to insist on it, I look forward to seeing you start a thread on this in the ATM forum.

Thanks for the recommendation, I'll look into Cantor's diagonal proof of uncountable sets a bit.

Ok you did explicitly mention it would take infinite time. Perhaps something can be rational and infinite at the same time but again i think we are mincing words a bit. Certainly saying that a solution takes infinite time to resolve is somewhat of a drawback!

DrRocket

2010-May-21, 05:30 AM

It most definitely means something my good sir! Which part did you not understand? Do you not know what a 4-vector is? Here is the wikipedia article: http://en.wikipedia.org/wiki/4-vector

I know perfectly well what a 4-vector is. Apparently you do not.

When you specify the 4-vector as you did as (0,0,0,-1) you have implicitly selected a coordinate frame. In that coordinate frame, you follow your 4-vector by setting back your clock. So, you apparently propose to find the center of the universe by going on daylight savings time. I doubt that such a strategy will work.

Second, the coordinate is observer-dependent. Any time-like 4-vector is (0,0,0,1) or (0,0,0,-1) in some reference frame. So there is no uniqueness to the vector that you think you have specified.

Third, there is the little matter of selecting and origin for your coordinate system. Any point, any point at all in spacetime, can be selected. Spacetime is not a vector space, even in the flat case it is an affine space.

Fourth, the presumption that you can specify a trajectory by specifying a vector requires that spacetime be flat. Spacetime is not flat. If it were there would be no such thing as gravity. It may be flat on the largest scales, but it most certainly not on ordinary scales -- like the scale of the galaxy or local group. This not only points out that you cannot use a simple vector to specify direction, but reinforces the problem with selection of an origin for your coordinate system.

Your problem is that you are speaking gibberish and you don't have the background to recognize that fact. You need to learn some mathematics and some relativity. Don't just provide a link to Wiki. Try actually reading the article. And maybe a book or two.

If you would care to provide some indication of your background in mathematics and physics I can perhaps recommend some suitable books.

I know perfectly well what a 4-vector is. Apparently you do not.

When you specify the 4-vector as you did as (0,0,0,-1) you have implicitly selected a coordinate frame. In that coordinate frame, you follow your 4-vector by setting back your clock. So, you apparently propose to find the center of the universe by going on daylight savings time. I doubt that such a strategy will work.

Second, the coordinate is observer-dependent. Any time-like 4-vector is (0,0,0,1) or (0,0,0,-1) in some reference frame. So there is no uniqueness to the vector that you think you have specified.

Third, there is the little matter of selecting and origin for your coordinate system. Any point, any point at all in spacetime, can be selected. Spacetime is not a vector space, even in the flat case it is an affine space.

Fourth, the presumption that you can specify a trajectory by specifying a vector requires that spacetime be flat. Spacetime is not flat. If it were there would be no such thing as gravity. It may be flat on the largest scales, but it most certainly not on ordinary scales -- like the scale of the galaxy or local group. This not only points out that you cannot use a simple vector to specify direction, but reinforces the problem with selection of an origin for your coordinate system.

Your problem is that you are speaking gibberish and you don't have the background to recognize that fact. You need to learn some mathematics and some relativity. Don't just provide a link to Wiki. Try actually reading the article. And maybe a book or two.

If you would care to provide some indication of your background in mathematics and physics I can perhaps recommend some suitable books.

CaptainToonces

2010-May-21, 05:32 AM

Not if you pencil them in, but if you insert millimeters in, sure it will get longer.

You're right, but taking the set of integers and adding in 2.5 and 3.5 doesn't add length to the set, it adds precision. Taking a number line of integers and adding in 3.5 doesn't make 1 any farther away from 4.

You're right, but taking the set of integers and adding in 2.5 and 3.5 doesn't add length to the set, it adds precision. Taking a number line of integers and adding in 3.5 doesn't make 1 any farther away from 4.

Geo Kaplan

2010-May-21, 05:35 AM

Thanks for the recommendation, I'll look into Cantor's diagonal proof of uncountable sets a bit.

Ok you did explicitly mention it would take infinite time. Perhaps something can be rational and infinite at the same time but again i think we are mincing words a bit. Certainly saying that a solution takes infinite time to resolve is somewhat of a drawback!

It's a drawback from a practical, human-lifetime point of view. But that doesn't make it nonsense.

Ok you did explicitly mention it would take infinite time. Perhaps something can be rational and infinite at the same time but again i think we are mincing words a bit. Certainly saying that a solution takes infinite time to resolve is somewhat of a drawback!

It's a drawback from a practical, human-lifetime point of view. But that doesn't make it nonsense.

DrRocket

2010-May-21, 05:47 AM

Don't move goalposts -- please read what I wrote with the same care I put into writing it. Note, for example, that I explicitly acknowledged that it would take infinite time. I said that there is nothing about this per se that makes it nonsense. It just makes it infinite. There's a difference. Again, just because it is nonsensical to you personally does not make it universally nonsensical. You have overlooked the possibility that the problem lies not with the concept, but with your own understanding.

I really do recommend that you study what Hilbert said about his "paradox", especially given that you cited it specifically. If your mind is truly open, you would also benefit from reading up on what Cantor achieved. Like you, his contemporaries had great difficulties placing concepts of infinity on a rational basis. Cantor showed that this could be done. It takes getting used to, I grant you that, but don't just reject the concept outright because it doesn't make immediate sense to you personally.

ETA: You seem to be laboring under the misapprehension that a program that doesn't execute in finite time is somehow indicative of an underlying irrationality. That is itself an ATM claim, so if you wish to insist on it, I look forward to seeing you start a thread on this in the ATM forum.

There is really no mention of time in the statement of the Hilbert Hotel example.

In any case it need not take infinite time at all. If each guest simply leaves his room, call it number N and moves to room 2N, the only even numbered rooms are occupied and all odd numbered rooms are empty. One has thereby, in one step, opened up a countably infinite number of rooms.

The Cantor diagonalization argument in its simplest form is used to show that it is impossible to make a one-to-one correspondence between the natural numbers and the real numbers. Thus the real numbers are uncountably infinite while the natural numbers, the integers, and rational numbers are all countably infinite. This is the beginning of the theory of cardinal numbers. There are differents "sizes" of infinity.

I really do recommend that you study what Hilbert said about his "paradox", especially given that you cited it specifically. If your mind is truly open, you would also benefit from reading up on what Cantor achieved. Like you, his contemporaries had great difficulties placing concepts of infinity on a rational basis. Cantor showed that this could be done. It takes getting used to, I grant you that, but don't just reject the concept outright because it doesn't make immediate sense to you personally.

ETA: You seem to be laboring under the misapprehension that a program that doesn't execute in finite time is somehow indicative of an underlying irrationality. That is itself an ATM claim, so if you wish to insist on it, I look forward to seeing you start a thread on this in the ATM forum.

There is really no mention of time in the statement of the Hilbert Hotel example.

In any case it need not take infinite time at all. If each guest simply leaves his room, call it number N and moves to room 2N, the only even numbered rooms are occupied and all odd numbered rooms are empty. One has thereby, in one step, opened up a countably infinite number of rooms.

The Cantor diagonalization argument in its simplest form is used to show that it is impossible to make a one-to-one correspondence between the natural numbers and the real numbers. Thus the real numbers are uncountably infinite while the natural numbers, the integers, and rational numbers are all countably infinite. This is the beginning of the theory of cardinal numbers. There are differents "sizes" of infinity.

DrRocket

2010-May-21, 05:49 AM

I am an expert on the value of the square root of -1.

I doubt that very much.

Do you mean in the context of the complex numbers, the quaternions, the hypercomplex numbers, operator theory, or something else ?

I doubt that very much.

Do you mean in the context of the complex numbers, the quaternions, the hypercomplex numbers, operator theory, or something else ?

Jens

2010-May-21, 05:54 AM

You're right, but taking the set of integers and adding in 2.5 and 3.5 doesn't add length to the set, it adds precision. Taking a number line of integers and adding in 3.5 doesn't make 1 any farther away from 4.

What do you mean by the "length" of the set? Do you mean to say there aren't twice as many numbers in the new set?

What do you mean by the "length" of the set? Do you mean to say there aren't twice as many numbers in the new set?

Geo Kaplan

2010-May-21, 05:54 AM

There is really no mention of time in the statement of the Hilbert Hotel example.

In any case it need not take infinite time at all. If each guest simply leaves his room, call it number N and moves to room 2N, the only even numbered rooms are occupied and all odd numbered rooms are empty. One has thereby, in one step, opened up a countably infinite number of rooms.

You are absolutely correct, of course.

In any case it need not take infinite time at all. If each guest simply leaves his room, call it number N and moves to room 2N, the only even numbered rooms are occupied and all odd numbered rooms are empty. One has thereby, in one step, opened up a countably infinite number of rooms.

You are absolutely correct, of course.

DrRocket

2010-May-21, 06:06 AM

You are absolutely correct, of course.

There is also another way to skin that cat, that serves to illustrate some things that happen when you deal with countable infinities.

Let each guest in room N move to room 2N, but do it sequentially. Let him do the move in 1/(2^N) seconds, then the total time required for the move is just the sum of the infinite geometric series, which in this case is 1 second. Some of the guests will have to be pretty quick packers and movers.

You can do the same thing with each guest moving one room down, opening up just one one room, but allowing each guest to notify the next that a move is needed, eliminating any need to coordinate simultaneous room switches.

There is also another way to skin that cat, that serves to illustrate some things that happen when you deal with countable infinities.

Let each guest in room N move to room 2N, but do it sequentially. Let him do the move in 1/(2^N) seconds, then the total time required for the move is just the sum of the infinite geometric series, which in this case is 1 second. Some of the guests will have to be pretty quick packers and movers.

You can do the same thing with each guest moving one room down, opening up just one one room, but allowing each guest to notify the next that a move is needed, eliminating any need to coordinate simultaneous room switches.

CaptainToonces

2010-May-21, 06:14 AM

When you specify the 4-vector as you did as (0,0,0,-1) you have implicitly selected a coordinate frame. In that coordinate frame, you follow your 4-vector by setting back your clock. So, you apparently propose to find the center of the universe by going on daylight savings time.

Setting back my clock? What are you talking about? I do not propose to travel back in time by setting back my clock, sir! As a 3-dimensional being I do have trouble looking in 4 dimensions, so I cannot really look in the direction (-1,0,0,0) but I can think about the past. The frame of reference is me--which follows naturally but you seem to have quite the trouble with it.

Second, the coordinate is observer-dependent. Any time-like 4-vector is (0,0,0,1) or (0,0,0,-1) in some reference frame. So there is no uniqueness to the vector that you think you have specified.

Again, the reference frame is me. To be most exact about me and my frame of reference, we can say that "me" is located at the tip of my nose. All things at rest to me are travelling in the direction (1,0,0,0) in my frame of reference. The 4-vector (-1,0,0,0) simply describes the opposite of that direction in my reference frame.

Third, there is the little matter of selecting and origin for your coordinate system. Any point, any point at all in spacetime, can be selected. Spacetime is not a vector space, even in the flat case it is an affine space.

The origin of the coordinate system has it's origin at the tip of my nose. It is further oriented by the symmetry of my face. And this frame of reference is sufficiently non-affine for my purposes of looking in the direction (-1,0,0,0)

Fourth, the presumption that you can specify a trajectory by specifying a vector requires that spacetime be flat. Spacetime is not flat. If it were there would be no such thing as gravity. It may be flat on the largest scales, but it most certainly not on ordinary scales -- like the scale of the galaxy or local group. This not only points out that you cannot use a simple vector to specify direction, but reinforces the problem with selection of an origin for your coordinate system.

The fact that spacetime may be curved does not mean that vectors can't be used to specify direction. That is just plain wrong. What are you going to use to specify directions besides vectors?

Your problem is that you are speaking gibberish and you don't have the background to recognize that fact. You need to learn some mathematics and some relativity. Don't just provide a link to Wiki.

And your problem is you're a jerk. Congratulations!

Setting back my clock? What are you talking about? I do not propose to travel back in time by setting back my clock, sir! As a 3-dimensional being I do have trouble looking in 4 dimensions, so I cannot really look in the direction (-1,0,0,0) but I can think about the past. The frame of reference is me--which follows naturally but you seem to have quite the trouble with it.

Second, the coordinate is observer-dependent. Any time-like 4-vector is (0,0,0,1) or (0,0,0,-1) in some reference frame. So there is no uniqueness to the vector that you think you have specified.

Again, the reference frame is me. To be most exact about me and my frame of reference, we can say that "me" is located at the tip of my nose. All things at rest to me are travelling in the direction (1,0,0,0) in my frame of reference. The 4-vector (-1,0,0,0) simply describes the opposite of that direction in my reference frame.

Third, there is the little matter of selecting and origin for your coordinate system. Any point, any point at all in spacetime, can be selected. Spacetime is not a vector space, even in the flat case it is an affine space.

The origin of the coordinate system has it's origin at the tip of my nose. It is further oriented by the symmetry of my face. And this frame of reference is sufficiently non-affine for my purposes of looking in the direction (-1,0,0,0)

Fourth, the presumption that you can specify a trajectory by specifying a vector requires that spacetime be flat. Spacetime is not flat. If it were there would be no such thing as gravity. It may be flat on the largest scales, but it most certainly not on ordinary scales -- like the scale of the galaxy or local group. This not only points out that you cannot use a simple vector to specify direction, but reinforces the problem with selection of an origin for your coordinate system.

The fact that spacetime may be curved does not mean that vectors can't be used to specify direction. That is just plain wrong. What are you going to use to specify directions besides vectors?

Your problem is that you are speaking gibberish and you don't have the background to recognize that fact. You need to learn some mathematics and some relativity. Don't just provide a link to Wiki.

And your problem is you're a jerk. Congratulations!

CaptainToonces

2010-May-21, 06:28 AM

There is really no mention of time in the statement of the Hilbert Hotel example.

In any case it need not take infinite time at all. If each guest simply leaves his room, call it number N and moves to room 2N, the only even numbered rooms are occupied and all odd numbered rooms are empty. One has thereby, in one step, opened up a countably infinite number of rooms.

Changing your algorithm from N + 1 to 2N does nothing to affect the infinite loop aspect of it. Think of writing a computer simulation, it still has to go through every natural number and multiply it by 2, which is an infinite amount of iterations.

In any case it need not take infinite time at all. If each guest simply leaves his room, call it number N and moves to room 2N, the only even numbered rooms are occupied and all odd numbered rooms are empty. One has thereby, in one step, opened up a countably infinite number of rooms.

Changing your algorithm from N + 1 to 2N does nothing to affect the infinite loop aspect of it. Think of writing a computer simulation, it still has to go through every natural number and multiply it by 2, which is an infinite amount of iterations.

Geo Kaplan

2010-May-21, 06:38 AM

Changing your algorithm from N + 1 to 2N does nothing to affect the infinite loop aspect of it. Think of writing a computer simulation, it still has to go through every natural number and multiply it by 2, which is an infinite amount of iterations.

My original answer embedded the same implicit assumption that you are making, namely, that we are simulating the process with a computer possessing finite memory, wordlength, etc., forcing one to serialize the process. DrRocket has properly pointed out that if we carry out the procedure as Hilbert posed it, it can be performed in non-infinite (indeed, asymptotically zero) time. Think of it as the result of exploiting infinite parallelism.

My original answer embedded the same implicit assumption that you are making, namely, that we are simulating the process with a computer possessing finite memory, wordlength, etc., forcing one to serialize the process. DrRocket has properly pointed out that if we carry out the procedure as Hilbert posed it, it can be performed in non-infinite (indeed, asymptotically zero) time. Think of it as the result of exploiting infinite parallelism.

DrRocket

2010-May-21, 06:42 AM

Changing your algorithm from N + 1 to 2N does nothing to affect the infinite loop aspect of it. Think of writing a computer simulation, it still has to go through every natural number and multiply it by 2, which is an infinite amount of iterations.

Wrong

Wrong

DrRocket

2010-May-21, 06:43 AM

And your problem is you're a jerk. Congratulations!

You know if you would sit back and think a bit you might learn something.

It turns to be much easier to learn with your mouth shut.

You know if you would sit back and think a bit you might learn something.

It turns to be much easier to learn with your mouth shut.

DrRocket

2010-May-21, 06:51 AM

Again, the reference frame is me. To be most exact about me and my frame of reference, we can say that "me" is located at the tip of my nose. All things at rest to me are travelling in the direction (1,0,0,0) in my frame of reference. The 4-vector (-1,0,0,0) simply describes the opposite of that direction in my reference frame.

Yep. And you proposed to locate the center of the universe by traveling in that direction. Which is going back in time. So, perhaps your proposal is to go back to the big bang. That does seem to address the spirit of the question.

The origin of the coordinate system has it's origin at the tip of my nose. It is further oriented by the symmetry of my face. And this frame of reference is sufficiently non-affine for my purposes of looking in the direction (-1,0,0,0)

The fact that spacetime may be curved does not mean that vectors can't be used to specify direction. That is just plain wrong. What are you going to use to specify directions besides vectors?

I wold probably try to specify a trajectory, perhaps with a vector field, which is quite a different thing.

Traveling over a smooth manifold is not the same thing as following a straight line. That is why when airplanes go to an eastern direction they do not just fly east. The fly a great circle route.

Your problem is that you think you know what you are talking about when in fact you are totally clueless. You need to pay attention to what you are being told. The mathematics is apparently quite beyond you at this point. Learn a bit.

And your problem is you're a jerk. Congratulations!

Nice touch.

Yep. And you proposed to locate the center of the universe by traveling in that direction. Which is going back in time. So, perhaps your proposal is to go back to the big bang. That does seem to address the spirit of the question.

The origin of the coordinate system has it's origin at the tip of my nose. It is further oriented by the symmetry of my face. And this frame of reference is sufficiently non-affine for my purposes of looking in the direction (-1,0,0,0)

The fact that spacetime may be curved does not mean that vectors can't be used to specify direction. That is just plain wrong. What are you going to use to specify directions besides vectors?

I wold probably try to specify a trajectory, perhaps with a vector field, which is quite a different thing.

Traveling over a smooth manifold is not the same thing as following a straight line. That is why when airplanes go to an eastern direction they do not just fly east. The fly a great circle route.

Your problem is that you think you know what you are talking about when in fact you are totally clueless. You need to pay attention to what you are being told. The mathematics is apparently quite beyond you at this point. Learn a bit.

And your problem is you're a jerk. Congratulations!

Nice touch.

tusenfem

2010-May-21, 08:14 AM

It most definitely means something my good sir! Which part did you not understand? Do you not know what a 4-vector is? Here is the wikipedia article: http://en.wikipedia.org/wiki/4-vector

Okay guys, that is enough! Discuss the topic and not the poster. Stop the snarky remarks right now or this thread will be closed and infractions will be handed out evenly over the snarky posters.

Okay guys, that is enough! Discuss the topic and not the poster. Stop the snarky remarks right now or this thread will be closed and infractions will be handed out evenly over the snarky posters.

CaptainToonces

2010-May-21, 08:16 AM

My original answer embedded the same implicit assumption that you are making, namely, that we are simulating the process with a computer possessing finite memory, wordlength, etc., forcing one to serialize the process. DrRocket has properly pointed out that if we carry out the procedure as Hilbert posed it, it can be performed in non-infinite (indeed, asymptotically zero) time. Think of it as the result of exploiting infinite parallelism.

I mean, you can substitute infinite parallelism for infinite time, but you've still required something infinite, which is rather hard to procure, be it time or processors.

I mean, you can substitute infinite parallelism for infinite time, but you've still required something infinite, which is rather hard to procure, be it time or processors.

CaptainToonces

2010-May-21, 08:23 AM

Yep. And you proposed to locate the center of the universe by traveliing in that direction.

No... i said nothing of travelling in that direction. I only proposed to ascertain in which direction it might lie... so that we might look at it or orient our view of the universe with respect to it.

No... i said nothing of travelling in that direction. I only proposed to ascertain in which direction it might lie... so that we might look at it or orient our view of the universe with respect to it.

DrRocket

2010-May-21, 08:29 AM

No... i said nothing of travelling in that direction. I only proposed to ascertain in which direction it might lie... so that we might look at it or orient our view of the universe with respect to it.

And just how do you propose to look along a timelike vector ?

And just how do you propose to look along a timelike vector ?

DrRocket

2010-May-21, 08:31 AM

I mean, you can substitute infinite parallelism for infinite time, but you've still required something infinite, which is rather hard to procure, be it time or processors.

The universe is not a computer. Hilbert's example has nothing to do with computers either. Computers are irrelevant to this discussion. Completely irrelevant.

The universe is not a computer. Hilbert's example has nothing to do with computers either. Computers are irrelevant to this discussion. Completely irrelevant.

tusenfem

2010-May-21, 08:37 AM

And your problem is you're a jerk. Congratulations!

Okay, and that costs you an infraction.

Congratulations

Okay, and that costs you an infraction.

Congratulations

Strange

2010-May-21, 09:26 AM

What do you mean by the "length" of the set? Do you mean to say there aren't twice as many numbers in the new set?

Well, no there aren't. If you have the infinite set of all natural numbers and add to it (say by doubling all the numbers and then filling the gaps) you end up with a set of the same infinite size. On the other hand, the set of real numbers is also infinite, but larger than the set of natural numbers.

[Given the real mathematicians around, I hope I haven't got that too wrong!]

Well, no there aren't. If you have the infinite set of all natural numbers and add to it (say by doubling all the numbers and then filling the gaps) you end up with a set of the same infinite size. On the other hand, the set of real numbers is also infinite, but larger than the set of natural numbers.

[Given the real mathematicians around, I hope I haven't got that too wrong!]

Strange

2010-May-21, 09:30 AM

I suppose that haggling about the nature of infinity is rather pointless since it is an imaginary concept.

It is a very well-defined concept. There are also different infinities. An infinite number of them, in fact...

In my understanding of the meanings of infinite and expand, it would seem that if you described something, say a number line, with the word "infinite" you would be implying that this thing is already expanded to the absolute maximum size that it can be expanded to and therefore cannot expand any more.

Which is where your understanding seems to fall short.

The Hilbert paradox proves only the nonsensical nature of trying to describe real things using the concept of infinity. You can move guest 1 to room 2, guest 2 to room 3, and so on infinitely, but that is an unclosed loop in your algorithm and if you did this in reality you would never get done ushering these poor guests out of their rooms. If you try to close the loop in the algorithm, you are left with a guest out of a room somewhere down the line.

It is not a "paradox" or "nonsensical", it is a demonstration of the nature of infinity.

It is not an "unclosed loop" or an "algorithm". You are thinking in mechanical or computing terms, which are irrelevant.

ETA: I see Dr. R has beaten me on that one.

It is a very well-defined concept. There are also different infinities. An infinite number of them, in fact...

In my understanding of the meanings of infinite and expand, it would seem that if you described something, say a number line, with the word "infinite" you would be implying that this thing is already expanded to the absolute maximum size that it can be expanded to and therefore cannot expand any more.

Which is where your understanding seems to fall short.

The Hilbert paradox proves only the nonsensical nature of trying to describe real things using the concept of infinity. You can move guest 1 to room 2, guest 2 to room 3, and so on infinitely, but that is an unclosed loop in your algorithm and if you did this in reality you would never get done ushering these poor guests out of their rooms. If you try to close the loop in the algorithm, you are left with a guest out of a room somewhere down the line.

It is not a "paradox" or "nonsensical", it is a demonstration of the nature of infinity.

It is not an "unclosed loop" or an "algorithm". You are thinking in mechanical or computing terms, which are irrelevant.

ETA: I see Dr. R has beaten me on that one.

captain swoop

2010-May-21, 09:36 AM

Maybe we need a reminder of the rules for the Q&A Forum.

Ask questions and get Mainstream answers. IF you don't understand ask more questions. If you don't agree then don't start making your own ATM claims in this forum. Start a thread in the ATM Forum.

Ask questions and get Mainstream answers. IF you don't understand ask more questions. If you don't agree then don't start making your own ATM claims in this forum. Start a thread in the ATM Forum.

astromark

2010-May-21, 11:18 AM

It is important to have a understanding of language to be part of this and these conversations...

Tolerance and understanding of those whom do not share your perception is not a rule.

By the behavior above maybe it could be... considered.

I have and continue to learn from what I gain from these conversations. It does disappoint me to see this descend as such... and why... ?

We all understand infinity. Any person that consumes themselves into astronomy must have a perception of what finite and, infinite are.

I am not the wizard of mathematics here and would argue that I do not need to be. I have a UNDERSTANDING and soon get guided if I need it from those whom will share. That the mainstream scientific view is endorsed and encouraged is very important.

On this question of the direction of the center of mass of the whole universe. I still do not understand what this discussion has become.

Some of what we claim to know may later be found to be wrong. That science strives to disprove every and all is the way forward. That is the way of science. I do not need to have a understanding of higher maths to know that the most distant objects yet observed are receding away at an alarming pace. Finding a edge is out of the question... If I can not define a edge the center point does not exist. Where would you put it. Now if that was all a wast of your time, consider this. Look at a full stop. It could be called a point. If you were a sub atomic particle that point is massive. Whats the point of this ? My loaf of bread is ready. If you can not see it then maybe thats not for you. I may not be understood well. I do imagine infinite. I may not know where the center is. Infinity may well be in every other direction.

Tolerance and understanding of those whom do not share your perception is not a rule.

By the behavior above maybe it could be... considered.

I have and continue to learn from what I gain from these conversations. It does disappoint me to see this descend as such... and why... ?

We all understand infinity. Any person that consumes themselves into astronomy must have a perception of what finite and, infinite are.

I am not the wizard of mathematics here and would argue that I do not need to be. I have a UNDERSTANDING and soon get guided if I need it from those whom will share. That the mainstream scientific view is endorsed and encouraged is very important.

On this question of the direction of the center of mass of the whole universe. I still do not understand what this discussion has become.

Some of what we claim to know may later be found to be wrong. That science strives to disprove every and all is the way forward. That is the way of science. I do not need to have a understanding of higher maths to know that the most distant objects yet observed are receding away at an alarming pace. Finding a edge is out of the question... If I can not define a edge the center point does not exist. Where would you put it. Now if that was all a wast of your time, consider this. Look at a full stop. It could be called a point. If you were a sub atomic particle that point is massive. Whats the point of this ? My loaf of bread is ready. If you can not see it then maybe thats not for you. I may not be understood well. I do imagine infinite. I may not know where the center is. Infinity may well be in every other direction.

DrRocket

2010-May-21, 03:02 PM

Well, no there aren't. If you have the infinite set of all natural numbers and add to it (say by doubling all the numbers and then filling the gaps) you end up with a set of the same infinite size. On the other hand, the set of real numbers is also infinite, but larger than the set of natural numbers.

[Given the real mathematicians around, I hope I haven't got that too wrong!]

You got it pretty much right.

The natural numbers are countably infinite. If you multiply by two you get the even natural numbers that are also of the same size (the one-to-one correspondence between the even numbers and the full set of natural numbers is obtained by just dividing by 2). To "fill in the gaps" in this case is just to add back in the odd numbers, so you wind up with the naturals again.

If you throw in the negative numbers you also get a countable infinite set, which is pretty easy to see from what you already observed. Just take your correspondence of the natural numbers with the even natural numbers, and then link the negative numbers to the odd natural numbers, leaving 0 along and you see that the integers are countably infinite.

The situation is more interesting with the rationals. They are also countable. The argument goes by showing that ordered pairs of integers (think numerator and denominator) are countable with a little more work. So, the natural numbers, the integers and the rational numbers are also countable.

Now think about "filling in the gaps". This is a bit more difficult that the gaps that you filled in with the natural numbers. The rationals have "holes" in them. Those "holes" are the irrational numbers, and they are pretty much everywhere, as we Will see. To fill them in you go through a topological process called "completion". There are couple of ways to do this. The classical way is to use Dedekind cuts, which are sets of rational numbers. One then defines an algebra of these sets and and order relation on them. The result is the real numbers. It takes a lot of work and is pretty boring, but effective. The other way is to use a more general construction that works for arbitrary "metric spaces" and uses equivalence classes of Cauchy sequences, which also takes some work to understand and explain. But the result is again the real numbers as you know them When you have done this, you have the real number line as you were taught it in high school.

An interesting thing happened along the way to the real numbers. You now have this big infinite set. It is what is called "complete" -- it doesn't have any holes. That is what makes calculus work. But you also have a new size of infinity. Cantor showed, with his famous diagonalization technique, that it is impossible to put the real numbers in one-to-one correspondence with the natural numbers. The real numbers are, in a precise sense, "bigger" than the natural numbers, the integers and the rational numbers. So we find that there are at least 2 different sizes of infinity. The complex numbers are just ordered pairs of real numbers (the real and imaginary parts) and are the same cardinality ("cardinality" is the technical term for size of a set) as the real numbers. Most, not all but most, of the mathematical objects that one encounters in mathematics and physics (I can't think of any counterexamples in physics at the moment) have cardinality no larger than that of the real numbers.

Given any set you can form the set of all subsets of that set. That new set always has cardinality larger than the original set. So there is no largest cardinal number. Cardinal numbers are themselves infinite.

If you take the natural numbers, or any countable infinite set and form the set of all subsets, you get a larger set. It can be shown that the cardinality of that set is the same as the cardinality of the real numbers.

Now, you might ask, is there a set of size in between that of the natural numbers and the real numbers ? The answer is interesting. The question has a name -- the "continuum hypothesis". Paul Cohen proved that this question is independent of the other usual axioms of set theory. So the answer is that one cannot prove it is either true or false from the usual foundations of mathematics. What is usually done in practice is to accept neither the continuum hypothesis nor its negation, although a few theorems, a very few, have been proved under the explicit assumption that the continuum hypothesis is true. Those theorems are not part of "mainstream" mathematics, but they neither are they "ATM" in the sense that the term is used on BAUT. They are just in a different class -- theorems with dependence on the continuum hypothesis What you never see is a mathematician using the continuum hypothesis, or any theorem dependent on it, without explicit comment -- such results are clearly labeled, and "quarantined".

If you are interested in this general sort of thing there is a nice, short, concise treatment in the book Naive Set Theory, by Paul Halmos. It s readily accessible to anyone with an interest in mathematics and doesn't take much background. All it takes is the interest and sufficient mathematical maturity to be able to follow the reasoning in the proofs.

[Given the real mathematicians around, I hope I haven't got that too wrong!]

You got it pretty much right.

The natural numbers are countably infinite. If you multiply by two you get the even natural numbers that are also of the same size (the one-to-one correspondence between the even numbers and the full set of natural numbers is obtained by just dividing by 2). To "fill in the gaps" in this case is just to add back in the odd numbers, so you wind up with the naturals again.

If you throw in the negative numbers you also get a countable infinite set, which is pretty easy to see from what you already observed. Just take your correspondence of the natural numbers with the even natural numbers, and then link the negative numbers to the odd natural numbers, leaving 0 along and you see that the integers are countably infinite.

The situation is more interesting with the rationals. They are also countable. The argument goes by showing that ordered pairs of integers (think numerator and denominator) are countable with a little more work. So, the natural numbers, the integers and the rational numbers are also countable.

Now think about "filling in the gaps". This is a bit more difficult that the gaps that you filled in with the natural numbers. The rationals have "holes" in them. Those "holes" are the irrational numbers, and they are pretty much everywhere, as we Will see. To fill them in you go through a topological process called "completion". There are couple of ways to do this. The classical way is to use Dedekind cuts, which are sets of rational numbers. One then defines an algebra of these sets and and order relation on them. The result is the real numbers. It takes a lot of work and is pretty boring, but effective. The other way is to use a more general construction that works for arbitrary "metric spaces" and uses equivalence classes of Cauchy sequences, which also takes some work to understand and explain. But the result is again the real numbers as you know them When you have done this, you have the real number line as you were taught it in high school.

An interesting thing happened along the way to the real numbers. You now have this big infinite set. It is what is called "complete" -- it doesn't have any holes. That is what makes calculus work. But you also have a new size of infinity. Cantor showed, with his famous diagonalization technique, that it is impossible to put the real numbers in one-to-one correspondence with the natural numbers. The real numbers are, in a precise sense, "bigger" than the natural numbers, the integers and the rational numbers. So we find that there are at least 2 different sizes of infinity. The complex numbers are just ordered pairs of real numbers (the real and imaginary parts) and are the same cardinality ("cardinality" is the technical term for size of a set) as the real numbers. Most, not all but most, of the mathematical objects that one encounters in mathematics and physics (I can't think of any counterexamples in physics at the moment) have cardinality no larger than that of the real numbers.

Given any set you can form the set of all subsets of that set. That new set always has cardinality larger than the original set. So there is no largest cardinal number. Cardinal numbers are themselves infinite.

If you take the natural numbers, or any countable infinite set and form the set of all subsets, you get a larger set. It can be shown that the cardinality of that set is the same as the cardinality of the real numbers.

Now, you might ask, is there a set of size in between that of the natural numbers and the real numbers ? The answer is interesting. The question has a name -- the "continuum hypothesis". Paul Cohen proved that this question is independent of the other usual axioms of set theory. So the answer is that one cannot prove it is either true or false from the usual foundations of mathematics. What is usually done in practice is to accept neither the continuum hypothesis nor its negation, although a few theorems, a very few, have been proved under the explicit assumption that the continuum hypothesis is true. Those theorems are not part of "mainstream" mathematics, but they neither are they "ATM" in the sense that the term is used on BAUT. They are just in a different class -- theorems with dependence on the continuum hypothesis What you never see is a mathematician using the continuum hypothesis, or any theorem dependent on it, without explicit comment -- such results are clearly labeled, and "quarantined".

If you are interested in this general sort of thing there is a nice, short, concise treatment in the book Naive Set Theory, by Paul Halmos. It s readily accessible to anyone with an interest in mathematics and doesn't take much background. All it takes is the interest and sufficient mathematical maturity to be able to follow the reasoning in the proofs.

Strange

2010-May-21, 03:18 PM

Thanks for a very nice summary. You lost me for a moment along the way, but this:

It is what is called "complete" -- it doesn't have any holes.

Is a nice insight. As is this:

That is what makes calculus work.

And the best short explanation of the continuum hypothesis I have seen.

If you are interested in this general sort of thing there is a nice, short, concise treatment in the book Naive Set Theory, by Paul Halmos. It s readily accessible to anyone with an interest in mathematics and doesn't take much background. All it takes is the interest and sufficient mathematical maturity to be able to follow the reasoning in the proofs.

I will look out for that.

It is what is called "complete" -- it doesn't have any holes.

Is a nice insight. As is this:

That is what makes calculus work.

And the best short explanation of the continuum hypothesis I have seen.

If you are interested in this general sort of thing there is a nice, short, concise treatment in the book Naive Set Theory, by Paul Halmos. It s readily accessible to anyone with an interest in mathematics and doesn't take much background. All it takes is the interest and sufficient mathematical maturity to be able to follow the reasoning in the proofs.

I will look out for that.

DrRocket

2010-May-21, 03:31 PM

I will look out for that.

You ought to be able to find it in a library. Or there is a used book in "fine" (which basically means like new) condition listed on the alibris.com site for $18.00 + $3.99 shipping at the moment. Use code "giving" and get a buck off.

It is the best little book on the subject of which I know. It is also pretty thin -- you can read it over a long weekend.

Halmos was one of the better writers of mathematics, and a pretty good mathematican. He wrote several other books, and all of them are very good, pretty much classics. His book on Hilbert spaces is a bit more advanced, but still the best introduction around. His book Finite Dimensional Vector Spaces is also the best treatment for analysts and physicists (not so much for algebraists).

You ought to be able to find it in a library. Or there is a used book in "fine" (which basically means like new) condition listed on the alibris.com site for $18.00 + $3.99 shipping at the moment. Use code "giving" and get a buck off.

It is the best little book on the subject of which I know. It is also pretty thin -- you can read it over a long weekend.

Halmos was one of the better writers of mathematics, and a pretty good mathematican. He wrote several other books, and all of them are very good, pretty much classics. His book on Hilbert spaces is a bit more advanced, but still the best introduction around. His book Finite Dimensional Vector Spaces is also the best treatment for analysts and physicists (not so much for algebraists).

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