I Is Infinity Possible Within Our Finite Universe?

[pelletboy]

Einstein mentioned that our universe if a finite spherical universe inside an infinite space. If this said infinite space is, as said, infinite, then infinity is possible? How do you explain the science of infinity?

 

[phinds]

Einstein mentioned that our universe if a finite spherical universe inside an infinite space. If this said infinite space is, as said, infinite, then infinity is possible? How do you explain the science of infinity?

I doubt Einstein said that since it is absolutely not in accordance with modern cosmology. There is no "outside". The universe is everything there is. It may be infinite in extent or it may not. Infinite seems to be the general consensus these days but it is not a known fact. If it is finite, the topology is not known.

 

[tade]

How do you explain the science of infinity?

Sorry, this question is not clear, you need to elaborate.

 

[russ_watters]

How do you explain the science of infinity?

Infinity is a mathematical concept, not a scientific one.

 

[sysprog]

Infinity is a mathematical concept, not a scientific one.

Infinity and related words refer to a set of conceptual abstractions which need not be restricted to mathematics. The question whether physical space is finite or infinite is not resolvable as simply a misapplication of the terms. It's a reasonable formulation of a question the correct answer to which is not known to be known.

 

[Buzz Bloom]

​

It's a reasonable formulation of a question the correct answer to which is not known to be known.

Hi sysprog:

I wonder if "the correct answer" is not only not known but also not scientifically knowable. I cannot imagine a scientific method that would result in certainty that that the universe was in fact infinite, or in fact finite. However, I think there are methods that can produce an approximation of the probability that the universe is infinite or is finite.

I understand that the universe model that best fits the currently available astronomical data gives a value for the average spatial curvature that is close to zero on the side that corresponds to a hyperbolic (infinite) spatial geometry. This value also comes with a range of error. From these values (and assumptions about the probability distribution) one can calculate the probability that the geometry is not hyperspherical, which means it is not finite.

I apologize for not citing a well known reference on this topic, and the numerical values, but I do not now have the time to look it up.

ADDED
https://arxiv.org/pdf/1502.01589.pdfFrom abstract

The spatial curvature of our Universe is found to be very close to zero, with|Ω_K|<0.005.​

6.2.4 Curvature

(49) The combined constraint shows impressive consistency with aflat universe: Ω_K=−0.005+0.016−0.017(95%,PlanckTT+lowP+lensing).​

I confess I do not know the correct method for working with an with different value for + and -. I am making a guess that since the two values are close, if I use the average 0.0165 as standard deviation and assume a Gaussian distribution, then the probability will approximately be

P(Ω_K > 0) ~= erf(0.005/0.0165) = ~erf(0.303) = ~0.332.​

Thus, based on the calculations of this reference, the probability is ~1/3 the universe is finite and ~`2/3 it is infinite.

Regards,
Buzz

 

[ZapperZ]

There are "infinities" in the mathematics everywhere that describe our world. This exists even in the electronics and materials that you use everyday (look up van Hove singularity in the density of states of material). In fact, if you look at ordinary conductors, the concept that produces Ohm's law requires that the single-particle spectral function be a delta function!

So why is everyone fixated only on "space" and such exotica?

Zz.

 

[sysprog]

I wonder if "the correct answer" is not only not known but also not scientifically knowable. I cannot imagine a scientific method that would result in certainty that that the universe was in fact infinite, or in fact finite.

It seems to me that proof of spatial finiteness in a spatially finite universe would be more likely to be attainable than would be proof of spatial infiniteness in a spatially infinite universe; if space is bounded, then we could imagine observing effects indicative of the existence of the boundary, but if space is unbounded, our not observing anything indicative of the existence of a boundary would not establish that there isn't one.

 

[sysprog]

There are "infinities" in the mathematics everywhere that describe our world. This exists even in the electronics and materials that you use everyday (look up van Hove singularity in the density of states of material). In fact, if you look at ordinary conductors, the concept that produces Ohm's law requires that the single-particle spectral function be a delta function!

So why is everyone fixated only on "space" and such exotica?

Zz.

We also don't know whether time and space at any scale are ultimately continuous or discrete.

 

[Buzz Bloom]

So why is everyone fixated only on "space" and such exotica?

Hi ZapperZ:

What causes my "fixation" is that I find it difficult to conceptualize about the geometry of the universe as being uncertain with respect to being finite or infinite. I am unaware of any other scientific question that has this particular uncertainty.

BTW, I added a reference and numbers to my previous post.

 

[anorlunda]

What causes my "fixation" is that I find it difficult to conceptualize

There are plenty of examples of physics that are so unlike our daily experience that we can never conceptualize them. Evolution did not wire our brains to conceptualize everything However we can understand them via the language of mathematics.

 

[ZapperZ]

Hi ZapperZ:

What causes my "fixation" is that I find it difficult to conceptualize about the geometry of the universe as being uncertain with respect to being finite or infinite. I am unaware of any other scientific question that has this particular uncertainty.

BTW, I added a reference and numbers to my previous post.

But you are not the OP, who questioned the "science of infinity". I pointed out that one doesn't need to look at issues of "space time" to discover that "infinities" exist almost everywhere in the mathematics of our physics. So do you and the OP also have no problems in accepting those infinities in your conductors and semiconductors, but only have problems with your concept of the "geometry of the universe"?

I find this very puzzling. People seem to think that such "exotic" properties only occurs in "exotic" physics, without realizing that the very things they use everyday exhibit similar properties.

Zz.

 

[Buzz Bloom]

... proof of spatial finiteness in a spatially finite universe would be more likely to be attainable than would be proof of spatial infiniteness in a spatially infinite universe; if space is bounded, then ...

I believe you have a misunderstanding about finiteness. The surface of the Earth is finite but it has no boundaries. A finite universe is similar except that the finite space is three dimensional rather than two.

Regards,
Buzz

 

[ZapperZ]

We also don't know whether time and space at any scale are ultimately continuous or discrete.

That has no connection with what I wrote. I don't need to know such thing to show you that there are infinities in the mathematics that describe the semiconductors that you are using to write this on your modern electronic devices.

Zz.

 

[sysprog]

I believe you have a misunderstanding about finiteness. The surface of the Earth is finite but it has no boundaries. A finite universe is similar except that the finite space is three dimensional rather than two.

Regards,
Buzz

In the sense in which I was using the terms 'bounded' and 'boundary', the volume of the Earth is bounded, and the surface of the Earth is its boundary.

 

[Buzz Bloom]

There are plenty of examples of physics that are so unlike our daily experience that we can never conceptualize them.

Hi anorlunda:

I agree with the above quote, but you missed the critical point to my reason for "fixation". It is the uniqueness of this particular uncertainty. Do you know of any other physical context in which it is uncertain about infinite or finite? (Please do not use QM interpretations as an example. I have given up on trying to conceptualize in that context.)

Regards,
Buzz

 

[jbriggs444]

In the sense in which I was using the terms 'bounded' and 'boundary', the volume of the Earth is bounded, and the surface of the Earth is its boundary.

But the point stands. Looking for evidence of finiteness by looking for a boundary is misguided. Finite but unbounded spaces (such as the surface of the earth) exist. We can never prove that the Earth's surface is finite by looking for an edge.

What we could do would be to get out our surveying equipment and look for triangles with internal angles that add to more than 180 degrees. Or we could just sail around the world.

 

[sysprog]

That has no connection with what I wrote. I don't need to know such thing to show you that there are infinities in the mathematics that describe the semiconductors that you are using to write this on your modern electronic devices.

Zz.

It's common knowledge that there are infinities in the mathematics that describe physical phenomena. We don't know whether anything physical is actually infinite. Your post questioned why there appeared to be a fixation on extension of space as to whether it is finite or infinite, and I presented an example of similarly unknown matters on the smallness scale as distinguished from the largeness scale; just as we don't know whether the universe is infinitely large, we don't know whether distances or durations can be infinitely or infinitesimally small.

 

[Buzz Bloom]

In the sense in which I was using the terms 'bounded' and 'boundary', the volume of the Earth is bounded, and the surface of the Earth is its boundary.

Hi sysprog:

I apologize for my failure to be clearer. The volume of the Earth is not analogous to the volume of a finite universe. However, the surface is a two dimensional analogue of the three dimensional hyper-surface of a four dimetional sphere. The interior volume of the four dimensional hypersphere is not part of the universe. It is only the three dimensional hyper-surface which approximates the shape of a finite universe based on a general relativity (GR) model.

Regards,
Buzz

 

[Buzz Bloom]

It's common knowledge that there are infinities in the mathematics that describe physical phenomena.

Hi sysprog:

I would appreciate seeing an example of such physical infinities. I am guessing there may be another misunderstanding that I may be able to explain.

Regards,
Buzz

 

[sysprog]

It's common knowledge that there are infinities in the mathematics that describe physical phenomena.

Hi sysprog:

I would appreciate seeing an example of such physical infinities. I am guessing there may be another misunderstanding that I may be able to explain.

Regards,
Buzz

What is the highest possible frequency, or shortest possible wavelength, of light? We can say mathematically that as frequency goes to 0, wavelength goes to infinity, and vice versa, but we don't know what physically is the shortest or longest possible time or distance.

 

[russ_watters]

Infinity and related words refer to a set of conceptual abstractions which need not be restricted to mathematics. The question whether physical space is finite or infinite is not resolvable as simply a misapplication of the terms. It's a reasonable formulation of a question the correct answer to which is not known to be known.

I agree that the specific question about the universe is in the realm of physics, but the one I responded do, as worded, seemed broader. We've had similar discussions such as "does infinity exist in the real world?" IMO, it is a useful descriptive tool, but the question leads to more problems than answers and is better left go.

 

[russ_watters]

We also don't know whether time and space at any scale are ultimately continuous or discrete.

Right. This is a commonly argued example that I just find so unnecessary. Many/most models assume infinitely divisible - whatever - but I see no real value in a debate over whether, for example, the infinitenumber of points on a ruler is physically "real". More often than not, this leads to misunderstandings such as Zeno's paradox.

 

[hutchphd]

Einstein mentioned that our universe if a finite spherical universe inside an infinite space. If this said infinite space is, as said, infinite, then infinity is possible? How do you explain the science of infinity?

It would be remiss to not mention here the name Georg Cantor who showed there are infinitely many kinds of infinity. He has pre-worried about some of this for you. Wrap your head around that...here's a start:
https://www.britannica.com/science/transfinite-number
His primary works are relatively approachable without too much pain.

 

[sysprog]

The volume of the Earth is not analogous to the volume of a finite universe.

It could be if we regard the space-distortional effects of mass as part of what is on a universal scale a mere local phenomena set. We don't know, for example, whether the universe itself is a finite or infinite Euclidean or non-Euclidean space, inside which everything so far observable to us is a mere speck.

 

[Buzz Bloom]

What is the highest possible frequency, or shortest possible wavelength, of light? We can say mathematically that as frequency goes to 0, wavelength goes to infinity, and vice versa, but we don't know what physically is the shortest or longest possible time or distance.

Hi sysprog:

I think the confusion is between (a) the absence of a limit, or (b) having a infinite value for a property. In the universe example, the size (e.g volume) is either infinite or it is a finite value (possibly changing over time). For a photon, its frequency and its wavelength are never infinite (and also never zero). Do you get the distinction?

Regards,
Buzz

 

[russ_watters]

I would appreciate seeing an example of such physical infinities. I am guessing there may be another misunderstanding that I may be able to explain.

The basic point of calculus is dealing with continuous change by incorporating infinities/infinitessimals. It would be hard to find a non-steady physical process that doesn't have to deal with infinity in its modeling.

Infinity is often regarded as way too exotic/mysterious and we're seeing a lot of that in this thread. That was largely the point of my first post.

 

[Buzz Bloom]

The basic point of calculus is dealing with continuous change by incorporating infinities/infinitessimals. It would be hard to find a non-steady physical process that doesn't have to deal with infinity.

Hi russ:

I get that the math deals with infinities. From many discussions here on the PF it has been made clear that in a small space in which some point has an infinite value for some property, such a point is called a singularity, and within a small space around the singularity it is said that the mathematics does not apply to the physics. It is in this sense that I am trying to make the point that aside from the possibly infinite size of the universe, there are no possible infinity values for physical properties.

Regards,
Buzz

 

[russ_watters]

...I am trying to make the point that aside from the possibly infinite size of the universe, there are no possible infinity values for physical properties.

I would think any property that has a zero value can have an inverse property with an infinite value, even if not always named/defined/useful. E.g., resistivity/conductivity.

 

[Buzz Bloom]

We don't know, for example, whether the universe itself is a finite or infinite Euclidean or non-Euclidean space, inside which everything so far observable to us is a mere speck.

Hi sysprog:

I have underlined "know" in the quote. This is to highlight what I understand to be a fundamental concept regarding all of science. The point is that "know" is a misleading word to use about science. In practice it has a usage meaning that is quite different from what is commonly intended as its meaning. In science a "fact" is believed to be true with a high degree of confidence, but also with the understanding that it is possible that at some future time (multiply reconfirmed) an observation with show that the fact is not true, or at least is not completely true. One excellent example from history is Newtonian mechanics.

A commonly used example of this understanding is that when a value is measured, or calculated from other measurements, an error range is also given. (See for example the values in the paper I cited in post #6.) The value and its error range allows a reader to calculate a probability that the actual value is in some specified range. I did that calculation in post #6.

P(Ω_K > 0) ~= erf(0.005/0.0165) = ~erf(0.303) = ~0.332.

So, this is what I believe that I know at the present time.

Based on the data described in the cited document, I know that the probability is ~1/3 that the universe is finite and ~2/3 that it is infinite.​

From my perspective this is qualitatively and scientifically different than, "We don't know, for example, whether the universe itself is a finite or infinite."

Regards,
Buzz

 

[Buzz Bloom]

Any property that has a zero value has an inverse property with an infinite value. E.g., resistivity/conductivity.

Hi Russ:

Thank you. You make a good point. I understand that it is commonly said that at some very cold temperatures the resistivity of some material becomes zero. (I.e., superconductivity.) If that is true, then that is a good and useful example, and I will need to adjust my perspective. Just a few questions to clarify my understanding.

Is it "certain" that the the resistivity becomes zero rather than just a very small positive value.​

Is there any substance and condition with zero conductivity?​

Regards,
Buzz

 

[sysprog]

So, this is what I believe that I know at the present time.
Based on the data described in the cited document, I know that the probability is ~1/3 that the universe is finite and ~2/3 that it is infinite.
From my perspective this is qualitatively and scientifically different than, "We don't know, for example, whether the universe itself is a finite or infinite."

Fair enough; however, I wasn't using 'know' in the sense of epistemic certainty, either. In my opinion, if in future we know whether space as we experience it is or is not curved enough (and does or does not have other characteristics sufficient) to make it topologically closed, while that would be a very significant piece of new knowledge, and would weigh strongly in favor of a finite theory, would not ipso facto resolve the question of finiteness or infiniteness of space, in any of the senses of 'know' that you have referenced.

 

[sysprog]

Hi russ:

I get that the math deals with infinities. From many discussions here on the PF it has been made clear that in a small space in which some point has an infinite value for some property, such a point is called a singularity, and within a small space around the singularity it is said that the mathematics does not apply to the physics. It is in this sense that I am trying to make the point that aside from the possibly infinite size of the universe, there are no possible infinity values for physical properties.

Regards,
Buzz

We know that the speed of light is finite, and we also know with high precision what its speed is, but we don't know its maximum frequency, or even for sure that it has a maximum frequency, although I think that we may learn in future that there is in fact physically a finite maximum frequency, and minimum time interval, and minimum distance, and to some degree of precision what those physical limits are.

 

[russ_watters]

It it "certain" that the the resistivity becomes zero rather than just a very small positive value.

Is there any substance and condition with zero conductivity?​

I think it's exactly zero with superconductivity, but I'm not sure.

Another example: Light transmittance/absorbance. AKA Beer's law:
https://teaching.shu.ac.uk/hwb/chemistry/tutorials/molspec/beers1.htm

Incidentally, I've been using Beer's Law totally incorrectly.

 

[Nugatory]

Do you know of any other physical context in which it is uncertain about infinite or finite?

Conditions at $r=0$ in a Schwarzschild black hole? It's a reasonable conjecture that some other physics is involved at sufficiently small values of $r$ to keep everything finite, but that is certainly not observationally confirmed and there is no compelling candidate theory.

 

[Nugatory]

I think the confusion is between (a) the absence of a limit, or (b) having a infinite value for a property.

Much of the problem here is in the informal English-language term "having an infinite value", which is somewhere between too vague to reason accurately with and just plain meaningless. Infinity is not a value (unless you happen to be an IEEE 854 nerd - irrelevant here) so anyone who is using that phrase is necessarily saying something else... and this is one of the times when the mathematicians' insistence on rigor produces better results than the physicists' lawless coercion of the math.

 

[Buzz Bloom]

It's a reasonable conjecture that some other physics is involved at sufficiently small values of rr to keep everything finite, but that is certainly not observationally confirmed and there is no compelling candidate theory.

Hi Nugatory:

Have two questions regarding the quote.
1. Would it be correct to say that it is impossible to observe any phenomena inside the event horizon?
2. Is it theoretically possible for there to be a compelling theory? That is, what could make a theory about what goes on inside the event horizon compelling?

Regards,
Buzz

 

[jbriggs444]

1. Would it be correct to say that it is impossible to observe any phenomena inside the event horizon?

If you dive in, you can observe. You just can't report back.

 

[Buzz Bloom]

Much of the problem here is in the informal English-language term "having an infinite value", which is somewhere between too vague to reason accurately with and just plain meaningless. Infinity is not a value (unless you happen to be an IEEE 854 nerd - irrelevant here) so anyone who is using that phrase is necessarily saying something else... and this is one of the times when the mathematicians' insistence on rigor produces better results than the physicists' lawless coercion of the math.

Hi Nugatory:

In what way is the following assertion "too vague to reason accurately with and just plain meaningless"?
Assuming that (a) the GR models of the universe are correct, and (b) that future astronomical data results in a mathematically correct calculation that the average curvature of the universe is positive with a confidence of 99.99%, it would then be also be known with a confidence of 99.99% that the volume of the universe is a value that is an infinite number of cubic meters.

Regards,
Buzz

 

[Buzz Bloom]

If you dive in, you can observe. You just can't report back.

Hi jbriggs:

Thank you for the insight. I will change my question to take your observation into account.

Would it be correct to say that it is impossible to observe from the outside any phenomena inside the event horizon, and also from the inside any phenomena happening closer to the center of the black hole than you are?

Regards,
Buzz

 

[jbriggs444]

Hi Nugatory:

In what way is the following assertion "too vague to reason accurately with and just plain meaningless"?
Assuming that (a) the GR models of the universe are correct, and (b) that future astronomical data results in a mathematically correct calculation that the average curvature of the universe is positive with a confidence of 99.99%, it would then be also be known with a confidence of 99.99% that the volume of the universe is a value that is an infinite number of cubic meters.

Regards,
Buzz

Putting a mathematical hat on, I'd be more comfortable saying that we would have high confidence that there are subsets which have more than any specified finite volume. [We would have to nail down with some precision what sort of subsets we had in mind and what volume measure we'd use].

That avoids the need to treat "infinity" as a value.

 

[sysprog]

and this is one of the times when the mathematicians' insistence on rigor produces better results than the physicists' lawless coercion of the math.

[rant]
I think that mathematicians should accept some of the responsibility for non-rigor; for example, treating the infinitesimal as non-zero when it's among the infinity of infinitesimals being summed, and then treating it as exactly zero when it's being disregarded at the asymptote, is formally inconsistent, and leads to absurdities that offend both rigorous and non-rigorous understanding, such as saying that having zero probability is not the same thing as being impossible, but impossible things are among those which have zero probability, and insisting on using zero to designate both of those kinds of zero without acknowledging that there must be more than one meaning in mathematics for zero, that zero in fact means one of the numbers on the continuum from the negative infinitesimal to the positive infinitesimal inclusively, with what is ordinarily meant by zero being the midpoint of that interval, and what is meant by zero in the statement 'the probability of choosing a rational number at random from within any interval within the reals is zero', is actually some number greater than the midpoint of the interval from the negative infinitesimal to the positive infinitesimal, whereas the impossible has a probability which is not even infinitesimally greater than zero, and so has what may be called a 'strictly not greater than zero' probability.
[/rant]

 

[Buzz Bloom]

[We would have to nail down with some precision what sort of subsets we had in mind and what volume measure we'd use].

Hi jbriggs:

I may be mistaken, but it seems to me to be impossible to define any such subset without saying either (a) "the subset is infinite in some appropriately defined dimension, or (b) the length value of some dimension is greater than any finite number.

Would it be satisfactory to you saying, "The volume of a GR universe with positive curvature is greater than any finite volume"? This avoids "infinite" by using instead a definition for infinite.

Regards,
Buzz

 

[ZapperZ]

It's common knowledge that there are infinities in the mathematics that describe physical phenomena. We don't know whether anything physical is actually infinite. Your post questioned why there appeared to be a fixation on extension of space as to whether it is finite or infinite, and I presented an example of similarly unknown matters on the smallness scale as distinguished from the largeness scale; just as we don't know whether the universe is infinitely large, we don't know whether distances or durations can be infinitely or infinitesimally small.

Then this is an issue that is not falsifiable, and thus, a topic in philosophy. Have fun going around in circles!

Zz.

 

[Nugatory]

2. Is it theoretically possible for there to be a compelling theory? That is, what could make a theory about what goes on inside the event horizon compelling?

"Compelling" is a matter of opinion and aesthetic preference, but I'd take as an operational definition that a theory is compelling if only a crackpot would argue, in the absence of a well-reasoned alternative, that we shouldn't be applying it.

I personally would not hesitate to trust classical E&M (applied within a sufficiently small region in a free-fall frame) on either side of the event horizon... for the same reason that we were willing to trust it to work on the far side of the moon before the era of space-travel. True, it hasn't yet been tested in that domain, but only a total crackpot would have suggested that it wasn't applicable and in that sense the theory was compelling.

We don't have a theory that as convincingly extends to extreme densities and pressures... but there's no reason why such a thing couldn't be discovered tomorrow.

 

[sysprog]

Then this is an issue that is not falsifiable, and thus, a topic in philosophy. Have fun going around in circles!

Zz.

A theory that a particular something is infinite may well be falsifiable. We know that some things once theorized to be infinite are in fact finite, for example, the speed of light. Similarly, we might in future learn of a minimum distance, a minimum time interval, and a consequent maximum frequency of light. I think we may not ever be able to prove anything physical to be infinite; however, some things may long resist every effort to prove them finite.

 

[jbriggs444]

I may be mistaken, but it seems to me to be impossible to define any such subset without saying either (a) "the subset is infinite in some appropriately defined dimension, or (b) the length value of some dimension is greater than any finite number.

I think you have mistaken the notion I was describing. None of the subsets involved need be infinite.

I am suggesting something very much like a limit. We play a game. You specify a volume (one cubic meter, one cubic light year, one hundred megaparsecs cubed, a cube with each side the factorial of Graham's number meters in length or whatever). Then I say "yes, there is a subset of the universe with at least that volume".

"For every volume V > 0, there is a subset with a volume at least that great".

That avoids not just the need to speak of infinity but also the need to speak of a numerical volume of the universe.

Edit: The above is just about definitions. I make no claim whether the universe really is or is not infinite in this sense.

 

[Matt Benesi]

I would appreciate seeing an example of such physical infinities. I am guessing there may be another misunderstanding that I may be able to explain.

I find myself wondering how many directions an averaged magnetic field is oriented in as I spin a magnet (relative to something else).

 

[sysprog]

You specify a volume (one cubic meter, one cubic light year, one hundred megaparsecs cubed, a cube with each side the factorial of Graham's number meters in length or whatever). Then I say "yes, there is a subset of the universe with at least that volume".

"For every volume V > 0, there is a subset [of the $\mathbb R^3$ spatial universe] with a volume at least that great".

(bracketed referent added)

Clearly, if and only if the universe is infinite (not volume-bounded), you and the other player (the immortal versions) can forever keep playing this game. But if it's finite, you'll at some point be blocked from completing your next turn, unless the last volume specified by the other player and acceded to by you were to be the actual finite limit, at which point the game would be concluded, unless the size of the universe were to increase.

That avoids not just the need to speak of infinity but also the need to speak of a numerical volume of the universe.

I don't see how how it accomplishes either of these. The game progressively increases the lower volumetric bound of the hypothetical $\mathbb R^3$ spatial universe, but it doesn't eliminate the consideration of whether there is physically an upper bound to the volumetric size. In ordinary vernacular, if there is, the universe is finite, and if not, it isn't.

Edit: The above is just about definitions. I make no claim whether the universe really is or is not infinite in this sense.

Me neither.

 

[jbriggs444]

The game progressively increases

There is no progression. The question is which player has a winning strategy.