Actual Infinity in Reality?

[polymath257]

OK, so to be clear. There is no 'outside' of spacetime. There is an outside of any particular spatial cross section (any other spatial cross section). But the collection of *all* such cross sections is 'spacetime'.

And again, the best answer to what the universe is expanding into, even for finite space, is 'the future'.

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Hmmm...a finite space does NOT imply an outside. That may be one of the issues here.

So, suppose that space is curved. In practice, what that means is that whatever direction you set off, if you keep going you will eventually come back around to your starting point (no actual travel--stay on one spatial cross section). So, you decide to take off in the 'up' direction and travel for a few billion light years, you will come back around to the start from the 'south' direction. Same with East vs West, and any direction and its opposite.

The reason I say not to actually travel is the time aspect of such: motion requires time and the time it would take to go 'around' is enough that the size changes during the trip.

So, just take a specific spatial cross section and go off in one direction, staying on that cross section. Eventually, with a finite space, you will come back around.

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Also, don't confuse the 'totality of all existence' with the 'totality of space at this time'.

Ok, so what you're saying (in the case of a finite universe) is that 'the totality of space at a certain time' is finite in the sense that eventually if you "movelessly" go across its full length (or whatever) in one direction, you eventually reach the point where you first began "movelessly" going (so basically a loop). Sort of like the surface of a ball, but we need to think in a higher dimension instead. IOW, finite does not always imply "edge". Did I get you right?

Yes. Finite in volume only. No edge

[SteveII]

Okay, let's back up to your initial position and try another tack. I think the best sentence that sums up your position is: " So, to the best I can tell, the idea that the universe is temporally infinite is consistent with a B theory of time and with some models of cosmology. So, ultimately, it doesn't appear that the case that you can't have an actual infinite has been made."

Under standard cosmology models, the B Theory of time has a beginning (at the time of the Bib Bang). An event creates a spacetime manifold. That at least make sense because we have a beginning and we are posit a potential infinite off into the future. By assuming an eternal universe model is correct, you assumed an eternal manifold and then...an actual infinite is possible.

After reading about the cosmology model you mentioned (CCC-Penrose). I noticed from your wiki link that "Penrose's basic construction[5] is to connect a countable sequence of open Friedmann–Lemaître–Robertson–Walker metric (FLRW) spacetimes, each representing a big bang followed by an infinite future expansion." This does not seem to be using the B Theory of time's manifold and simply claiming it never had a beginning (therefore an actual infinite). The theory proposes a "countable sequence" of different spacetime manifolds. Each manifold exists in sequence and therefore was never part of one big spacetime that existed as one eternal block.

So, it would seem that proposing the two theories together does not get you to even a model of an actual infinite and brings the question right back to, metaphysically speaking, can we have an actual infinite of past events? Since you can't get an eternal spacetime block out of any theory, you must have successive states of affairs. If you have successive states of affairs, they cannot be past infinite, because you will never get to our current state of affairs because an infinite number of prior states of affairs would have to happen.

There are a few misunderstandings here. First, and trivially, events don't create spacetimes. Spacetimes are made out of events.

I think that is wrong. The standard model most ascribe to has a change in the singularity that resulted in the spacetime manifold we experience now. 

Second, the sequence of spacetimes in the Penrose model are each internal to the previous. Again, there is no beginning to the sequence and the 'overall' manifold is not one of the spacetimes, but a sort of multiverse with time going infinitely into the past.

And, yes, there was an infinite sequence into the past in this model, necessarily. So you are (again) wrong about the difficulty of an infinite past. And yes, an infinite number of prior states happened. So? 

Where is the contradiction to an infinite number of prior states having happened? If at each time, that is always the case, there is no contradiction.

To get around the problem I brought up, all of those sequential spacetime manifolds (see where I quoted) have to already exist. That does not seem to be part of the theory. Please address this problem specifically. Most of your statements just seem to be assertions and not part of the model.

[polymath257]

There are a few misunderstandings here. First, and trivially, events don't create spacetimes. Spacetimes are made out of events.

I think that is wrong. The standard model most ascribe to has a change in the singularity that resulted in the spacetime manifold we experience now. 

Second, the sequence of spacetimes in the Penrose model are each internal to the previous. Again, there is no beginning to the sequence and the 'overall' manifold is not one of the spacetimes, but a sort of multiverse with time going infinitely into the past.

And, yes, there was an infinite sequence into the past in this model, necessarily. So you are (again) wrong about the difficulty of an infinite past. And yes, an infinite number of prior states happened. So? 

Where is the contradiction to an infinite number of prior states having happened? If at each time, that is always the case, there is no contradiction.

To get around the problem I brought up, all of those sequential spacetime manifolds (see where I quoted) have to already exist. That does not seem to be part of the theory. Please address this problem specifically. Most of your statements just seem to be assertions and not part of the model.

First, the singularity is NOT an event: it is a failure of the coordinate system to describe a situation. In this case, if you use standard general relativity, it describes a limit of infinite curvature.

Yes, it actually is part of the theory that the *previous* spacetimes already exist.

[GrandizerII]

Ok, so what you're saying (in the case of a finite universe) is that 'the totality of space at a certain time' is finite in the sense that eventually if you "movelessly" go across its full length (or whatever) in one direction, you eventually reach the point where you first began "movelessly" going (so basically a loop). Sort of like the surface of a ball, but we need to think in a higher dimension instead. IOW, finite does not always imply "edge". Did I get you right?

Yes. Finite in volume only. No edge

Well, a lot of blood and sweat were dropped, but consider me now convinced you have a good point here. I've always been aware of such concept, but for some reason, this didn't cross my mind here in this thread.

[Jehanne]

I think that is wrong. The standard model most ascribe to has a change in the singularity that resulted in the spacetime manifold we experience now. 


To get around the problem I brought up, all of those sequential spacetime manifolds (see where I quoted) have to already exist. That does not seem to be part of the theory. Please address this problem specifically. Most of your statements just seem to be assertions and not part of the model.

First, the singularity is NOT an event: it is a failure of the coordinate system to describe a situation. In this case, if you use standard general relativity, it describes a limit of infinite curvature.

Yes, it actually is part of the theory that the *previous* spacetimes already exist.

Note -- A "singularity" is an actual infinite!!

[polymath257]

Yes. Finite in volume only. No edge

Well, a lot of blood and sweat were dropped, but consider me now convinced you have a good point here. I've always been aware of such concept, but for some reason, this didn't cross my mind here in this thread.

When a cosmologist talks of space being finite, this is  always what is meant. NOT that space has some edge. At least, I can't think of a counter-example.

[SteveII]

I just read your opening post. I don't know what has been presented.

You ask if something can exist "something concrete (not abstract)". -edit- that is infinite.

The theory of a Black-hole is just that. Is there an end to how dense a singularity can get? No. Theoretically it can continue to be more and more dense. At infinitude. There is nothing more "concrete" then matter its self. A black-hole is that mater condensed and can theoretically continue to condense more and more.

If you mean something that is proven beyond theory, then the idea of numbers. One can always count higher. There is no end to numbers. Count and count and count on. You'll always have another number to go to.

Ina black hole, you are talking about a property of being infinitely dense. That is not the same thing as having an infinite number of anything in real life.

Being able to count infinitely is a potential infinite. You can never actually arrive at infinity. The question the thread is concerned about is whether an actual infinite can exist (something real that has an infinite number of things).

[Anomalocaris]

First, the singularity is NOT an event: it is a failure of the coordinate system to describe a situation. In this case, if you use standard general relativity, it describes a limit of infinite curvature.

Yes, it actually is part of the theory that the *previous* spacetimes already exist.

Note -- A "singularity" is an actual infinite!!

If we have no other reason to suspect our theory that arrives at singularity is flawed, then we might think when the theory points to singularity, then an actual infinity is a probability to be taken seriously.

But in case of gravity and blackholes, we actually have strong reason to believe the theory of gravity breaks down well before things reach infinity.

[Jehanne]

Note -- A "singularity" is an actual infinite!!

If we have no other reason to suspect our theory that arrives at singularity is flawed, then we might think when the theory points to singularity, then an actual infinity is a probability to be taken seriously.

But in case of gravity and blackholes, we actually have strong reason to believe the theory of gravity breaks down well before things reach infinity.

WLC needs to stop quoting "singularity theorems" if he believes that actual infinities are a logical impossibility.  He's not only contradicting himself; he's talking out of his a-hole.

By the way, the Universe is not some giant 4-D (5-D, if you include time) spatial ball (the spatial dimensions being curved through some sort of hyperspace); if it was, then it would have a center of mass, and it doesn't.

[surreptitious57]

The lifespan of a photon in vacuum is regarded as being infinite though it would have to be predicated on two other factors
Namely that its journey is only ever through empty space and that the Universe itself is either temporally or spatially infinite