Actual Infinity in Reality?

[GrandizerII]

The Universe is growing in spatial extent, since, on an intergalactic scale, distances are increasing.  Even if the Universe is infinite in spatial extent, the distances between any two galaxies is finite, and so, one has Cantor's concept of "infinities within infinities":

https://blogs.scientificamerican.com/the...thematics/

A Universe that is an "actual infinite" in space and time is completely reasonable.

Yes to this.

Yes to a multiverse.

No to a finite universe expanding into nothing.

Glad you brought this up though because all the more reason to go with actual infinity rather than against.

[Anomalocaris]

No to a finite universe expanding into nothing.

Q. How does finite universe expanding into nothing differ from an infinite but non-homogenous universe in which one particular part is expanding?

I think we don’t really know whether our universe is on a cosmological scale truly homogenous or not. I think that for ease of conception, we use a working assumption of homogeneity, but not a fundamental assumption of homogeneity. We might say it is probably reasonably homogenous to a scale significantly larger than the currently observable universe because we detect no discernible edge effects out to the limits of our observation horizon. But I think we really don’t know it is homogenous, and it would actually pose questions of why and how we can’t answer and only dodge if the universe is truly homogenous over such a large distance scale as for there to have been no possible way for two ends of the scale to reach any sort of equilibrium at any time after the Big Bang.

[GrandizerII]

How does finite universe expanding into nothing differ from an infinite but non-homogenous universe in which one particular part is expanding?

Depends on how one is treating the word "nothing". If its to be taken to mean a literal nothingness, then in the first case this is somehow happening, and in the second case, instead of "nothingness", we have the wider universe itself.

[Anomalocaris]

How does finite universe expanding into nothing differ from an infinite but non-homogenous universe in which one particular part is expanding?

Depends on how one is treating the word "nothing". If its to be taken to mean a literal nothingness, then in the first case this is somehow happening, and in the second case, instead of "nothingness", we have the wider universe itself.

I think that’s a sort of semantic that is less useful than it appears. If what is “nothing” can be represented by the complete canceling out of all wave functions. Each component wave function can be said to be there, but for the presence of all the others which exactly cancel it out. So where everything cancels out, is there something or nothing?

[GrandizerII]

Depends on how one is treating the word "nothing". If its to be taken to mean a literal nothingness, then in the first case this is somehow happening, and in the second case, instead of "nothingness", we have the wider universe itself.

I think that’s a sort of semantic that is less useful than it appears. If what is “nothing” can be represented by the complete canceling out of all wave functions. Each component wave function can be said to be there, but for the presence of all the others which exactly cancel it out. So where everything cancels out, is there something or nothing?

I would still consider it a something if only because there it is existing in such a case, and it is a product of other things interacting and canceling each other out. Even if its pretty much just a void.

I see what you mean though. Its just that people have these absolute conception of nothing, so when scientists speak of such things as universe expanding into nothing or coming from nothing, then it comes off as absurd to some of us who fail to understand what they mean exactly.

[polymath257]

The universe is expanding, but not into a true literal nothing. That word "into" implies something. Otherwise, a logical interpretation of the sentence "the universe is expanding into nothing" would that the universe isnt expanding at all.

Sometimes physicists have to be careful with how they word things. No wonder why we get theists misrepresenting what many of us believe about the universe or cosmos.

The *best* answer to the question 'what is the universe expanding into?' is THE FUTURE.

The universe is expanding into the future.

And, while that may sound like a trick of language, there is a strict sense in which it is completely correct in general relativity.

So, as discussed above, spacetime is a four dimensional construct: it consists of all of space and all of time as a single geometric entity. Spatial cross sections of this entity (i.e, space) are 'larger' for later times than they are for earlier times. That is what is means to say space is expanding. It just means that a later 3 dimensional cross section is larger (in some sense) that an earlier one.

A very basic analogy may make this easier to comprehend. Suppose that the surface of the earth represents spacetime. In this analogy, think of higher latitudes as being later in time. Different longitudes are different spatial locations.

So, a 'time slice' is a cross section of the surface of the earth that has the same latitude: a latitude line. This represents space at a specific time.
 

Now, in the southern hemisphere, as we move north (which is later in time), the latitude lines get larger: space is expanding. In the northern hemisphere, as we move north, the latitude lines get smaller: space is contracting.

Also, we have a 'beginning' at the south pole and an 'end' at the north pole. In this analogy, time is finite with 'singularities' at the poles.

Now, what are the latitude lines expanding into? Different latitude lines aren't even in the 'same space' because they are at different times. They expand or contract into the future!

Now, this is a *very* simplistic analogy, but the essences are there: spacetime is a single geometric construct, space corresponds to cross sections of that geometry. Different times correspond to different cross sections, and expansion/contraction are determined by comparing two different cross sections.

[SteveII]

I set up four such thought experiments above. We clearly have contradictions that arise when comparing these possible worlds.

You are once again conflating the ability or inability to imagine something as being the same as demonstrating that something is or is not logically possible.  It is the strength of imagination that we have the ability to conceive of impossible things.  I can imagine that there is a possible world where God does not exist.  Have I thus demonstrated that God is not a necessary being?  No, I have not.  If God is a necessary being and I imagine that God does not exist in a possible world, all I've shown is that my imagination is at odds with my assumptions.  Your thought experiments don't add anything to the assumptions and conclusions you had prior to the thought experiment.

As long as we're on the subject though, allow me to make several notes,

1. Infinity, while treated as a number, is not a number in the sense that the counting numbers are.  Therefore the equations you are presenting above have to be construed as set theoretic operations.  As such, there is nothing contradictory about the set theoretic results.  It only appears that way if you are construing the equations as normal numerical operations.  Thus presenting the equations adds nothing and seems to serve only to mislead.

2. From what I understand of possible worlds semantics, the idea of comparing one possible world to another, different possible world is not supported.  If you think it is, then I'd request that you show which possible world semantics you are referencing.  If you can't compare possible worlds meaningfully, then attempting to even formulate Hilbert's hotel's operation in terms of possible world semantics is not possible.

3. Hilbert's hotel applies to sets that are countably infinite.  If time is continuous and infinite, it would seem that the set of all possible moments is uncountably infinite.  In that event, Hilbert's hotel simply wouldn't apply.  As long as we're throwing around burden of proof questions, I think you are obligated to either show that time is not continuous, or that even if it is, that the set of all possible moments is a countable infinity.  Otherwise, we can simply dispense with Hilbert's hotel, as it does not cover all the possibilities for a temporally infinite universe that I have raised.  An objection which only applies to some of the possibilities but not all cannot possibly demonstrate that all cases are impossible.

But the real problem is that your position is that all of them are true despite the obvious contradictions. You have not shown why we should accept the contradictions other than to wonder if that's just the way it is. It seems to me that you have some burden of proof to shoulder if you are proposing ignoring obvious contradictions. 

Since you haven't actually shown any such alleged contradictions, I have great difficulty making sense of your complaint here.  I'm supposed to refute the existence of contradictions you haven't demonstrated?  That's ballsy, but ridiculous.  I can't refute a case that you haven't made.  So, no, I don't assume any burden of proof to show that something you claim exists doesn't exist.  You need to first demonstrate the existence of these alleged contradictions.  Once you've shouldered your burden of proof, we'll see what obligations I have in return.

Why isn't that question begging? The proposition is that an actual infinity does not exist. To dismiss thought experiments on the basis they don't work with actual infinities needs a little more support.

Well, first of all, you're moving the goalpost.  The question is whether or not an actual infinity is logically possible.  The claim that the proposition is whether an actual infinity actually exists is asking me to demonstrate that a specific actual infinity is in fact actual.  Those are different standards.  I don't know that I could prove that time is temporally infinite even if I wanted to do so.  I never claimed as much.  Only that the idea of a temporal infinity is consistent, both logically, and with known models of physics and cosmology.  I believe I've done that.  Your job as my interlocutor is to show that I've missed a contradiction which exists.  In that context, I am suggesting that the so-called absurdity that results in the thought experiment may be a product of an incomplete set of intuitions about reference.  It's a possibility.  Your task, is to show that the absurdity in Hilbert's hotel is metaphysically real, not just a product of intuitional failure.  You so far have not done so, and continue to talk around the problem rather than addressing it.

It seems to me you are accepting an actual infinity as a brute fact. Can accepting a brute fact really be considered "logical".

No, as I just pointed out, I'm accepting that the hypothesis that time is temporally infinite is both logically and physically consistent.  But that doesn't seem to be your point here.  Your question as to whether what I'm doing is "logical" seems to be nothing more than a rhetorical smear.  If you're reduced to such smears, I have to question what you hope to achieve with it?  Treating something as a brute fact is neither logical nor illogical, so I can only assume that, instead, you are simply trying to suggest that I'm being irrational.  I don't see that as a productive path to a convincing argument.  It seems little more than an attempt to distract from the point I made, that you had not shown that any metaphysical assumption has been violated, and thereby avoid actually showing such a contradiction. 

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.

[polymath257]

No to a finite universe expanding into nothing.

Q.  How does finite universe expanding into nothing differ from an infinite but non-homogenous universe in which one particular part is expanding?

I think we don’t really know whether our universe is on a cosmological scale truly homogenous or not.  I think that for ease of conception, we use a working assumption of homogeneity, but not a fundamental assumption of homogeneity.   We might say it is probably reasonably homogenous to a scale significantly larger than the currently observable universe because we detect no discernible edge effects out to the limits of our observation horizon.    But I think we really don’t know it is homogenous, and it would actually pose questions of why and how we can’t answer and only dodge if the universe is truly homogenous over such a large distance scale as for there to have been no possible way for two ends of the scale to reach any sort of equilibrium at any time after the Big Bang.

Yes, this is a difficulty on the *very* large scale that includes parts o the universe have are not and never have been causally connected to us. Once we get outside of this sort of 'Hubble bubble', there is very little we can say because *by definition* it has no influence on what we see.

On the other hand, the *observed* homogeneity across the observable universe *was* an issue because of the very issue you raise: how is it possible to get such homogeneity when the distant parts were not causally connected?

The resolution of this (and some other) paradoxes was the inflationary model In this, there was a stage of *very* rapid expansion, so that all we see today *was* causally connected in the early universe and *that* was  when the homogeneity was formed. After about 80 successive doublings of the size of the universe, the particle producing this expansion decayed (it may have even been the HIgg's particle---jury still out on this one) to produce the 'ordinary' expansion we see today.

One obvious issue is that any inhomogeities are *so* far away that they don't have any measurable effects. Unfortunately, this means that we cannot tell the difference between a 'flat' space and one with a very small curvature (the curvature is spread out during inflation). So, if there is a *very* small negative spatial curvature, it is possible that space is finite. If the curvature is zero or negative, it is more likely that space is infinite.

We simply cannot tell. But there is no *logical* problem with space being infinite.

---

You are once again conflating the ability or inability to imagine something as being the same as demonstrating that something is or is not logically possible.  It is the strength of imagination that we have the ability to conceive of impossible things.  I can imagine that there is a possible world where God does not exist.  Have I thus demonstrated that God is not a necessary being?  No, I have not.  If God is a necessary being and I imagine that God does not exist in a possible world, all I've shown is that my imagination is at odds with my assumptions.  Your thought experiments don't add anything to the assumptions and conclusions you had prior to the thought experiment.

As long as we're on the subject though, allow me to make several notes,

1. Infinity, while treated as a number, is not a number in the sense that the counting numbers are.  Therefore the equations you are presenting above have to be construed as set theoretic operations.  As such, there is nothing contradictory about the set theoretic results.  It only appears that way if you are construing the equations as normal numerical operations.  Thus presenting the equations adds nothing and seems to serve only to mislead.

2. From what I understand of possible worlds semantics, the idea of comparing one possible world to another, different possible world is not supported.  If you think it is, then I'd request that you show which possible world semantics you are referencing.  If you can't compare possible worlds meaningfully, then attempting to even formulate Hilbert's hotel's operation in terms of possible world semantics is not possible.

3. Hilbert's hotel applies to sets that are countably infinite.  If time is continuous and infinite, it would seem that the set of all possible moments is uncountably infinite.  In that event, Hilbert's hotel simply wouldn't apply.  As long as we're throwing around burden of proof questions, I think you are obligated to either show that time is not continuous, or that even if it is, that the set of all possible moments is a countable infinity.  Otherwise, we can simply dispense with Hilbert's hotel, as it does not cover all the possibilities for a temporally infinite universe that I have raised.  An objection which only applies to some of the possibilities but not all cannot possibly demonstrate that all cases are impossible.


Since you haven't actually shown any such alleged contradictions, I have great difficulty making sense of your complaint here.  I'm supposed to refute the existence of contradictions you haven't demonstrated?  That's ballsy, but ridiculous.  I can't refute a case that you haven't made.  So, no, I don't assume any burden of proof to show that something you claim exists doesn't exist.  You need to first demonstrate the existence of these alleged contradictions.  Once you've shouldered your burden of proof, we'll see what obligations I have in return.


Well, first of all, you're moving the goalpost.  The question is whether or not an actual infinity is logically possible.  The claim that the proposition is whether an actual infinity actually exists is asking me to demonstrate that a specific actual infinity is in fact actual.  Those are different standards.  I don't know that I could prove that time is temporally infinite even if I wanted to do so.  I never claimed as much.  Only that the idea of a temporal infinity is consistent, both logically, and with known models of physics and cosmology.  I believe I've done that.  Your job as my interlocutor is to show that I've missed a contradiction which exists.  In that context, I am suggesting that the so-called absurdity that results in the thought experiment may be a product of an incomplete set of intuitions about reference.  It's a possibility.  Your task, is to show that the absurdity in Hilbert's hotel is metaphysically real, not just a product of intuitional failure.  You so far have not done so, and continue to talk around the problem rather than addressing it.


No, as I just pointed out, I'm accepting that the hypothesis that time is temporally infinite is both logically and physically consistent.  But that doesn't seem to be your point here.  Your question as to whether what I'm doing is "logical" seems to be nothing more than a rhetorical smear.  If you're reduced to such smears, I have to question what you hope to achieve with it?  Treating something as a brute fact is neither logical nor illogical, so I can only assume that, instead, you are simply trying to suggest that I'm being irrational.  I don't see that as a productive path to a convincing argument.  It seems little more than an attempt to distract from the point I made, that you had not shown that any metaphysical assumption has been violated, and thereby avoid actually showing such a contradiction. 

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.

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.

[GrandizerII]

The universe is expanding, but not into a true literal nothing. That word "into" implies something. Otherwise, a logical interpretation of the sentence "the universe is expanding into nothing" would that the universe isnt expanding at all.

Sometimes physicists have to be careful with how they word things. No wonder why we get theists misrepresenting what many of us believe about the universe or cosmos.

The *best* answer to the question 'what is the universe expanding into?' is THE FUTURE.

The universe is expanding into the future.

And, while that may sound like a trick of language, there is a strict sense in which it is completely correct in general relativity.

So, as discussed above, spacetime is a four dimensional construct: it consists of all of space and all of time as a single geometric entity. Spatial cross sections of this entity (i.e, space) are 'larger' for later times than they are for earlier times. That is what is means to say space is expanding. It just means that a later 3 dimensional cross section is larger (in some sense) that an earlier one.

A very basic analogy may make this easier to comprehend. Suppose that the surface of the earth represents spacetime. In this analogy, think of higher latitudes as being later in time. Different longitudes are different spatial locations.

So, a 'time slice' is a cross section of the surface of the earth that has the same latitude: a latitude line. This represents space at a specific time.
 

Now, in the southern hemisphere, as we move north (which is later in time), the latitude lines get larger: space is expanding. In the northern hemisphere, as we move north, the latitude lines get smaller: space is contracting.

Also, we have a 'beginning' at the south pole and an 'end' at the north pole. In this analogy, time is finite with 'singularities' at the poles.

Now, what are the latitude lines expanding into? Different latitude lines aren't even in the 'same space' because they are at different times. They expand or contract into the future!

Now, this is a *very* simplistic analogy, but the essences are there: spacetime is a single geometric construct, space corresponds to cross sections of that geometry. Different times correspond to different cross sections, and expansion/contraction are determined by comparing two different cross sections.

But then what about the earth itself and beyond? There is nevertheless still something beyond "finite space" in this analogy (which, as you say, is a simplistic one anyway). So even if we were like 2D beings who couldn't understand the notion of any sort of direction in the third dimension or whatever, and weren't aware that the "latitude timeslices" were placed in the context of a planet that is situated in an outer space, it does seem like even in our ignorance of what it is exactly, we still have to logically conclude there has to be something out there, as opposed to literally nothing.

Perhaps that's not what you were disputing, but I'm just saying it here just to let you know where I'm coming from here.

[SteveII]

Steve seems to have disappeared from the thread, either because he has realized he was wrong, or because he still thinks he’s right.  Or, I dunno; maybe he needed to do laundry or something.  😛

Actually sailing on a 46' sailboat for a week as part of sailing certification.