Where does idea of infinite size of universe come from?

[andrew s]

8 hours ago, saac said:

The language we use to define the circle (mathematics) produces what we define as infinity. Infinity is a consequence of that language it does not follow it exits in nature. 

Jim 

I am not sure what this means. What do you feel does exist in nature. 

Elections, em fields, quantum fields, energy and entropy are all components of our mathematical models of reality.

Chairs, tables and pebbles are objects of our naming and of our perceptual model of the world. 

The integers, reals and irrational are part of our  mathematics   

Which if any of these are part of reality? If not what is?

Regards Andrew 

 

[andrew s]

On 21/05/2019 at 23:13, vlaiv said:

However, there might be mathematical identity that is only true for infinite sums and diverges considerably when finite version is used, and such identity used to derive physical law.

Not quite this but fun for a Thursday morning.

Consider the sum 1+2 +3 +4 +5...     (where ... means keep going for as long as there are new integers to add - forever) 

If you try this on your calculator you will find it diverges (continues to get bigger) and does not converge to a specific number.

(Note the avoidance of the term infinity)

All well and good no objections so far I trust!

Now Riemann came up with a function now called the Riemann zeta function

zeta(s) = 1/1^s  + 1/2^s + 1/3^s ... which is defined for all values of s but for s = 1

Now if you put s = -1 you get zeta(-1) = -1/12  (try it here https://keisan.casio.com/exec/system/1180573439  as I doubt it is on you calculator )

All well and dandy. Now put s = -1 into the equation above for zeta(s) and you get

zeta(-1) = 1 + 2 + 3 + 4 + 5....

So the same sum both diverges and converges!

Does this matter is it just proving the unreality of Mathematics.

Well the first divergent sum is just counting (albeit forever). The second crops up in QED and the calculation of the correct strength of the Casimir force.

Zee in "Quantum Field Theory in a Nutshell" derives the answer without using the zeta function but there is no reason not to use it any more than not using sin, cos or tan.

Happy days.

Regards Andrew

PS in QED it is often the opposite of @vlaiv request in that you get excellent agreement with the first few term but it diverges when you add significantly more.

 

 

Edited May 23, 2019 by andrew s

[saac]

4 hours ago, andrew s said:

I am not sure what this means. What do you feel does exist in nature. 

Elections, em fields, quantum fields, energy and entropy are all components of our mathematical models of reality.

Chairs, tables and pebbles are objects of our naming and of our perceptual model of the world. 

The integers, reals and irrational are part of our  mathematics   

Which if any of these are part of reality? If not what is?

Regards Andrew 

 

Yes but those are more than the predictions of mathematics Andrew. While mathematics can use the concept of infinity as a useful operator it does not define what in reality exits. Consider that our laws of mathematics and physics no longer hold in the realm of infinity - the singularity whether it exists or not cannot be described by our mathematical laws of physics.  In a sense I accept infinity exists in one respect - as I 've said if our universe has a size then that is by definition infinity but in practice I do not believe it has the meaning that mathematics ascribes to it that which it holds in popular culture.  When it appears in our equations we generally take a sharp intake of breadth and say we have an incomplete understanding here as with the singularity. 

Jim 

 

Edited May 23, 2019 by saac

[andrew s]

25 minutes ago, saac said:

Yes but those are more than the predictions of mathematics Andrew. While mathematics can use the concept of infinity as a useful operator it does not define what in reality exits. Consider that our laws of mathematics and physics no longer hold in the realm of infinity - the singularity whether it exists or not cannot be described by our mathematical laws of physics.  In a sense I accept infinity exists in one respect - as I 've said if our universe has a size then that is by definition infinity but in practice I do not believe it has the meaning that mathematics ascribes to or tat which it holds in popular culture.  When it appears in our equations we generally take a sharp intake of breadth and say we have an incomplete understanding here as with the singularity. 

Jim 

 

Ok Jim no problem with that.

However, I want you to give me example of what does exist in nature.

This is so I can understand how you decide what does and doesn't exist in nature or is real however you prefer to put it.

Regards Andrew 

[vlaiv]

2 hours ago, andrew s said:

Not quite this but fun for a Thursday morning.

Consider the sum 1+2 +3 +4 +5...     (where ... means keep going for as long as there are new integers to add - forever) 

If you try this on your calculator you will find it diverges (continues to get bigger) and does not converge to a specific number.

(Note the avoidance of the term infinity)

All well and good no objections so far I trust!

Now Riemann came up with a function now called the Riemann zeta function

zeta(s) = 1/1^s  + 1/2^s + 1/3^s ... which is defined for all values of s but for s = 1

Now if you put s = -1 you get zeta(-1) = -1/12  (try it here https://keisan.casio.com/exec/system/1180573439  as I doubt it is on you calculator )

All well and dandy. Now put s = -1 into the equation above for zeta(s) and you get

zeta(-1) = 1 + 2 + 3 + 4 + 5....

So the same sum both diverges and converges!

Does this matter is it just proving the unreality of Mathematics.

Well the first divergent sum is just counting (albeit forever). The second crops up in QED and the calculation of the correct strength of the Casimir force.

Zee in "Quantum Field Theory in a Nutshell" derives the answer without using the zeta function but there is no reason not to use it any more than not using sin, cos or tan.

Happy days.

Regards Andrew

PS in QED it is often the opposite of @vlaiv request in that you get excellent agreement with the first few term but it diverges when you add significantly more.

 

 

That sum always diverges. What happens with Riemann zeta function is that we use something called analytical continuation to "calculate" value of zeta(-1). We replace original function with function that "behaves" in the same way in terms of derivatives and such and that function gives us answer zeta(-1) = -1/12

Here is a good brief article on subject:

https://plus.maths.org/content/infinity-or-just-112

Now, with respect to usage in QM - I think it is very indicative of what I've been writing previously - math that we use is approximation / model of what is truly going on. In this case we assumed that some term is sum of series in our theory while in reality it probably more acts as analytical continuation of such sum, and for this reason observations agree with what we get when we plug the number in Riemann zeta.

Just yesterday I was reading sort of "renormalization for dummies" kind of text - and while it is still out of my grasp, it did provide insight into similar understanding - we try different "hacks" to circumvent problems that arise from usage of math and concepts of infinities - because inherently underlying world does not work with infinities.

I'm going to quote author of the text as I found their remarks very intriguing:

Quote

So what happens if we start with a nonrenormalizable theory and play this "renormalization group" game? Our Lagrangian will typically have a bunch of terms in it: some nasty ones that are making the theory nonrenormalizable, and some nice ones that would give a renormalizable theory if we just threw out the nasty ones. Each of these terms is multiplied by a coupling constant. Now let's look at the corresponding physical coupling constants as we crank up the distance scale D'.

As we do this, the physical coupling constants in front of the nasty nonrenormalizable terms get smaller and smaller, approaching zero! At large distances, nonrenormalizable interactions become irrelevant!

This is an incredibly important fact, because it may explain why the quantum field theory that seems to describe our world — the Standard Model — is renormalizable. There may be all sorts of strange quantum gravity stuff going on at very short distance scales — perhaps spacetime is not even a continuum! But if at larger scales we assume that ordinary quantum field theory on flat spacetime is a reasonably accurate approximation to what's going on, then this renormalization group stuff assures us that at still larger scales, nonrenormalizable interactions are going to look very weak.

Emphasis in text was added by me to point out intriguing parts - fact that theory "approximates" real underlying reality and that we "dodge" certain cases where infinities lead without really explaining underlying physical reality - which might be one way or the other. Like I already mention fact that spacetime is considered continuous might be just "artifact" of mathematical approximation.

[saac]

1 hour ago, andrew s said:

Ok Jim no problem with that.

However, I want you to give me example of what does exist in nature.

This is so I can understand how you decide what does and doesn't exist in nature or is real however you prefer to put it.

Regards Andrew 

Ok I sense you are toying with me now so I will accept the challenge;  may I take an infinity of time to consider?   Didn't Einstein invoke a comparison of sitting on a hot stove  compared to a park bench with a pretty girl  to describe time.  That may have had to to with relativity of time I suppose.   

Ok so, I  exist as do you or say water or a rock  or light , heat, radio waves, or the electron.   I don't think any of those would require a physical infinity for their existence to hold. We may be able to describe each of their nature's using an expression of mathematics which in turn may or may not rely upon the operator we call infinity.    I think that is what  I am getting at regarding my difficulty in accepting a physical infinity !

Jim 

ps I can well understand why Cantor was driven mad by this although that may be fake news  

pps.  If infinity does have a physical nature would we be able to physically measure it to confirm its existence;  would information be able to span the infinity of whatever property it exits in?  Would that require an infinity of time?  I have no idea - arghhh   

Edited May 23, 2019 by saac

[saac]

13 hours ago, vlaiv said:

You have to be careful about usage of word infinite. One that contains "everything that exists" is not necessarily infinite. Infinite that we are talking about has very exact meaning and is related to concept of geometry of space. If for any given number, no matter how large you select it to be, you can find two points in space that have separation greater than that number in your chosen units - then universe is infinite in size.

Infinite universe in above sense has infinite amount of "stuff" in it if we assume homogeneity, or if we assume that density is finite number.

I accept that vlaiv .  But could not my infinite universe with infinite "stuff" not refer to the infinite number line I have used as my measuring stick.  The infinite stuff therein being the numbers. 

Is it possible for any infinite system to have an infinite value of one property while not another?  And what if the mathematical model links those two variables, if one is infinite what doe that imply of the other?  I honestly have no idea.

 

Jim 

[vlaiv]

15 minutes ago, saac said:

I accept that vlaiv .  But could not my infinite universe with infinite "stuff" not refer to the infinite number line I have used as my measuring stick.  The infinite stuff therein being the numbers. 

Is it possible for any infinite system to have an infinite value of one property while not another?  And what if the mathematical model links those two variables, if one is infinite what doe that imply of the other?  I honestly have no idea.

 

Jim 

Ok, yes, indeed as mathematical concept we can have something like bounded set with infinite number of elements. This is fairly simple to see - let's take simple case of set of real numbers between 0 and 1.

This set is bounded - there is upper limit and lower limit and there is a number which is greater than distance between any two set members - like 2, if we define distance to be difference of two numbers.

We can "apply" such a model to hypothetical universe, but we can easily see that such universe is "unphysical" - density is infinite, all "elements of reality" have only one property - value and there is infinite number of elements of reality that are in principle - "unknowable". Spatial extent of any element is infinitesimally small - or 0, and by "unknowable" I mean that there is a bunch of numbers - irrational numbers that can't be specified - one would need infinite amount of information to exactly specify them (infinite number of digits to be written down - which can only be done in infinite amount of time).

This is interesting game to play actually - we can observe another set of numbers that has similar premise but different properties - let's take only rational numbers in range of 0 to 1. This set is also bounded, there are infinite number of elements, there is no lower limit on how "close" two numbers can be, density is infinite, however this sort of set is "sparse".

We know this by virtue that on real line between 0 and 1 we have real numbers. Now if we split real numbers into two subsets - rational and irrational and remove irrational numbers - there must be some sort of "hole" left in that place and there is some sparseness in the set - and yet again - density is infinite and set elements are infinitesimally close

Mind bending stuff really

 

[Kev M]

7 hours ago, vlaiv said:

 - density is infinite and set elements are infinitesimally close

 

 

Isn't density a ratio.... and hence by definition....rational ?

 

 

[Kev M]

We have to be aware of infinities and there place in the real world...to take an example highlighted upon before and look at it differently.

There are obviously more non-integer numbers than integers.... I think this makes sense.

In fact there are infinitely more non integers than integers.... again look at 1 & 2 and count the non-integers between them ( might take a while ).

Statistically Integers do not exist !

It can have weird effects when you move from reality to imagination.

It would seem to make sense that infinities do not exist in the real world, ie nothing you can observe can be infinite. ( or be part of anything infinite ).

 

 

 

[vlaiv]

11 minutes ago, Kev M said:

Isn't density a ratio.... and hence by definition....rational ?

 

 

Point being? Density in this case is - take any interval within 0-1 range, 0.3-0.4 for example and divide length of that interval with count of numbers within that interval or 0.1 / ∞ ~ 0 (any finite number divided by infinity is infinitesimal).

5 minutes ago, Kev M said:

In fact there are infinitely more non integers than integers

This statement is not correct, and again one has to be careful when working with infinities. Basic division or types of infinities are countable infinity and uncountable infinity.

Interesting fact about rational numbers is that there is countable infinite number of them - same as integers - this means that there is "the same number" of rational numbers and integers.

Have a look here:

https://www.homeschoolmath.net/teaching/rational-numbers-countable.php

Page already emphasizes important sentence, and I'll just copy it here:

"a set is countable either if it s finite, or it is infinite and you can find a one-to-one correspondence between the elements of the set and the set of natural numbers"

Finding one to one correspondence between two sets in fact means they contain same number of elements (as for each element in one set there is exactly one corresponding element).

Real numbers on the other hand are uncountable - meaning there is vastly more real numbers than both integer numbers and rational numbers - again point given above, difference being all irrational numbers.

[andrew s]

Hi @vlaiv yes I know about analytical continuation etc. As I said a bit of fun !

Hi @saac playing only slightly. I am not trying to say infinity is real in the sense of you or I are  but for me it is as real as a googleplex or 0. Although it is not a number!

However,  I do find it difficult to know if the entities of our physical models are real or not. If you take an electron as an example do they exist in a solid where they have no well defined location or only when free or not at all in the sense a pebble does. I don't  know.

@vlaiv on renormalistaion that is an interesting quote but I do get tired of physicist saying all problems will dissolve when we "quantise" spacetime and gravity. I have a sneaking feeling we will not find a theory of everything and gravity and QM will remain unreconciled.(No evidence just a hunch. )

Regards Andrew 

 

 

Edited May 23, 2019 by andrew s

[Kev M]

11 hours ago, vlaiv said:

  11 hours ago, Kev M said:

Isn't density a ratio.... and hence by definition....rational ?

11 hours ago, vlaiv said:

Point being? 

If its rational it cant be Infinite......therefore you cant have infinite density !

 

 

[Kev M]

11 hours ago, vlaiv said:

11 hours ago, Kev M said:

In fact there are infinitely more non integers than integers

This statement is not correct, and again one has to be careful when working with infinities. 

True, but I think it shows a point.

The above example shows that if you treat infinity as a number it becomes nonsense.

Infinity is not a number... so you cant think of it as such, you cant perform mathematical operations with it ( its not a number ).

Hence the universe cant be infinite because there would be an "infinite number" of something ( maybe Apples... I like Apples ).

Can't have an "infinite number" as this is meaningless... might as well have a " red number" or a "fluffy number" again meaningless.

 

 

 

[andrew s]

1 hour ago, Kev M said:

True, but I think it shows a point.

The above example shows that if you treat infinity as a number it becomes nonsense.

Infinity is not a number... so you cant think of it as such, you cant perform mathematical operations with it ( its not a number ).

Hence the universe cant be infinite because there would be an "infinite number" of something ( maybe Apples... I like Apples ).

Can't have an "infinite number" as this is meaningless... might as well have a " red number" or a "fluffy number" again meaningless.

 

 

 

You are right infinity is not a number. You can, however, do mathematical manipulation on it. Not that they are particularly relevant to this discussion. 

I think as @vlaiv has pointed out you need to be careful what infinity means in this context. 

We are discussing the LCDM cosmological model in which spacetime is geometry. The geometry of Einstein's Gerneral Relativity. In this context a spatially flat universe has and always has had infinite spatial extent. All this  means is that you can continue in any spatial direction without bound. This is a totally respectable geometrical notion.

You and others may read more into this beyond the geometrical meaning. For example as@vlaiv does that this leads to multiple repetitions etc. These require logic and physical reasoning beyond LCMD. Perfectly legitimate but I tend to the view that they are often misguided.

Regards Andrew 

 

Edited May 24, 2019 by andrew s

[Alien 13]

59 minutes ago, Kev M said:

Can't have an "infinite number" as this is meaningless... might as well have a " red number" or a "fluffy number" again meaningless.

Its funny you mention this but for quite a few weeks after my stroke (I am fine now) I could smell numbers and it was very distinctive, looking at a digital clock at 7.30 am was an especially nice smell.

Alan

[andrew s]

13 minutes ago, Alien 13 said:

Its funny you mention this but for quite a few weeks after my stroke (I am fine now) I could smell numbers and it was very distinctive, looking at a digital clock at 7.30 am was an especially nice smell.

Alan

And at what time was the worst pong?

Regards Andrew 

Edited May 24, 2019 by andrew s

[Alien 13]

2 minutes ago, andrew s said:

And the what time was the worst pong?

Regards Andrew 

I cant remember specifics but my door number was not pleasant, a sort of burnt plastic/cucumber combination.

I dont however recall this odd effect in reverse i.e. smells producing numbers. I did look around on the net and this type of cross sensor anomalies is not uncommon in stroke victims and did make me wonder if our concept of Maths and numbers is actually hard wired into our brains. This might explain why the concepts of zero and infinity cause problems and why we struggle with using alternative reasoning to view the world.

Alan

[andrew s]

39 minutes ago, Alien 13 said:

I cant remember specifics but my door number was not pleasant, a sort of burnt plastic/cucumber combination.

I dont however recall this odd effect in reverse i.e. smells producing numbers. I did look around on the net and this type of cross sensor anomalies is not uncommon in stroke victims and did make me wonder if our concept of Maths and numbers is actually hard wired into our brains. This might explain why the concepts of zero and infinity cause problems and why we struggle with using alternative reasoning to view the world.

Alan

How we perceive the world is indeed very complex and in some cases quite anomalous to the norm as your experience confirms. This is why I teased @saac about what is real. Our perception may all differ to a more or less significant degree and what we naturally take as real might be quite different if evolution had equipped us differently. 

All the worlds a concept and we are just avatars of a boltzmann brain .🤔

Regards Andrew 

[vlaiv]

2 hours ago, Kev M said:

If its rational it cant be Infinite......therefore you cant have infinite density ! 

 

 

I think you are mixing two things here - rational number and the way that rational number is "produced".

Rational number is a number, and it is produced by division of two numbers - specifically for rational number to be a number you need to divide whole number with a natural number (this definition avoids division by 0).

Density is ratio - it is "produced" in the same way as rational number, and for most part it is in fact rational number because of this. However there is no restriction on what should go into division here. You can divide infinity and divide with infinity.

Imagine infinite number line and finite number of elements on it. What is density of these finite elements? It is 0. How is it 0? Because ratio of finite number and infinite number is 0. Now imagine opposite - finite line segment and infinite number of elements within that line segment - like 0-1 line segment and real numbers on it. Here we have infinity / 1 = infinity. Density is here infinite and it's not rational number.

2 hours ago, andrew s said:

You and others may read more into this beyond the geometrical meaning. For example as@vlaiv does that this leads to multiple repetitions etc. These require logic and physical reasoning beyond LCMD. Perfectly legitimate but I tend to the view that they are often misguided.

I'd like to point out, with risk of it sounding that I'm again trying to put discussion back in certain direction, which I'm not - is that my conclusions stem from exactly what you pointed out - infinite spatial extent. This and basic assumption of cosmology - homogeneity, and the fact that we observe finite density in our observable patch indeed leads to those things that I mentioned.

Edited May 24, 2019 by vlaiv

[RolandKol]

"Where does idea of infinite size of universe come from?"

I think a bit simpler on this matter.

Once "we" started to understand we are the part of the universe, the obvious question came in, - there does it end?

As no one knows the answer, all possible versions been applied,  - model of three whales, Ptolemy and Aristotlel models, and a bit later "Infinite" came in.

Further more, if someone will want to support the statement "it is Not infinite", but in other shape/form witch is closed, - he will face more very interesting and intriguing questions, like what is beyond the border? What universe expands to? and etc
 

It is hard to imagine the Infinite Universe, at least for me... And not interesting at all to be honest!
It sounds like a "dead end" for cosmology... Plus, the infinite one!  

It eliminates all the other interesting questions and probably is comfy in some cases to apply.

[andrew s]

43 minutes ago, vlaiv said:

'd like to point out, with risk of it sounding that I'm again trying to put discussion back in certain direction, which I'm not - is that my conclusions stem from exactly what you pointed out - infinite spatial extent. This and basic assumption of cosmology - homogeneity, and the fact that we observe finite density in our observable patch indeed leads to those things that I mentioned.

I realise that and fully acknowledge it. However, you also apply an argument that in essence says that if you have an infinite state space then repetition is inevitable, there will be boltzmann brains etc. This is a common view and you will find many serious deployments of this argument by many much more capable than I.

I however, think it is wrong. I think the way probability is employed  in these arguments is in error similar to the error that leads to Olblers' paradox.  It is this area I am not motivated to discuss further. Not because I fear being proved wrong but it is just too complex for me to articulate clearly enough. 

Please forgive my reticence. 

Regards Andrew 

 

[vlaiv]

17 minutes ago, RolandKol said:

"Where does idea of infinite size of universe come from?"

I think a bit simpler on this matter.

Once "we" started to understand we are the part of the universe, the obvious question came in, - there does it end?

As no one knows the answer, all possible versions been applied,  - model of three whales, Ptolemy and Aristotlel models, and a bit later "Infinite" came in.

Further more, if someone will want to support the statement "it is Not infinite", but in other shape/form witch is closed, - he will face more very interesting and intriguing questions, like what is beyond the border? What universe expands to? and etc
 

It is hard to imagine the Infinite Universe, at least for me... And not interesting at all to be honest!
It sounds like a "dead end" for cosmology... Plus, the infinite one!  

It eliminates all the other interesting questions and probably is comfy in some cases to apply.

I guess infinite comes from age before we actually knew that there is finite limit on speed of light, GR was not developed and we had no idea that there were other galaxies out there. It was just starry night sky and eternal static universe.

And that is I guess quite ok, however we developed all these theories in the mean time, started developing cosmological model backed up by observations, and it turns out that this model provides three distinct solutions for curvature. I'm just wondering why no one said, hm wait, we no longer need to have something that is infinite in spatial extent - we just saw that time was not infinite and "eternal" part is out of the picture, static is also out of the picture as we know there is quite a bit of dynamic behavior of universe - but we decided to keep infinite part as main focus

Ok, I understand why it might be so - because of "flat" part and reluctance to give up notion of isotropy - which I believe is fair for the time being until there is evidence against it.

Btw, we do have very good understanding of bounded positive curve space and how it addresses questions you mentioned.

First off, there is no need for border in finite geometry, actually border hurts basic assumption - cosmological principle and we would need to abandon it to discuss bordered universe (points near the border are distinctly different than those further away). Simple way to visualize borderless universe is to just look at surface of the earth - you can move indefinitely around the surface of the earth - even following straight line and you will never reach the edge. Most people instinctively think: "but hold on, there is up and down", and that is because surface of the sphere is 2D geometry embedded in 3D geometry.

We can then say, OK, but does that mean that our 3D geometry needs to be embedded in 4D geometry? No, it does not need to be, and since there is no evidence of 4D spatial dimensions, I don't think we need to concern us with that notion at all (we don't need it, there is no evidence for it). 2D geometry can also be analyzed without need for 3D geometry - we just use 3D geometry because we are used to it.

But what is "outside" of this universe? And where does it expands into? Ok, these are a bit tricky, not in terms of answering, but rather in terms of getting a grasp on what answer means.

Answers are rather simple: there is nothing outside our universe - and here I don't mean it's empty, what I mean is: outside does not exist. Answer to second one is - universe is not expanding into something, universe is just expanding.

Here is a bit more elaborate explanation of answers, or rather a way you should be thinking about these answers to help you understand what they mean and why is it so.

First is outside bit - we can't get "outside" of universe as there is no border of it. We can't even get close to "outside" because again - there is no border. We don't have any evidence that there is outside as all we know and can interact with and can measure is within this universe. There simply is no outside. We are just used to thinking about outside because of our daily experiences - it always involves a border of some sorts and we are so accustomed to this way of thinking that we find it hard to accept that there could not be "outside" at all.

Similar thing happens with expanding. We are used to things expanding into something, so we can't mentally grasp that things can expand on their own. But expansion is increase of dimensions - dimensions can be external - if there is outside, and/or internal - distance between things. If there is no outside as a reference point, but distance between things increases - it is also expansion - just without into part as there is nothing to expand into.

[saac]

14 hours ago, andrew s said:

Hi @saac playing only slightly. I am not trying to say infinity is real in the sense of you or I are  but for me it is as real as a googleplex or 0. Although it is not a number!

However,  I do find it difficult to know if the entities of our physical models are real or not. If you take an electron as an example do they exist in a solid where they have no well defined location or only when free or not at all in the sense a pebble does. I don't  know.

Regards Andrew 

 

 

Hi Andrew absolutely nothing wrong with the playing it's been both educational and fun - I'm really enjoying this thread.   I think as you are finding out the trouble I have with infinity is not with the mathematical concept rather than an infinity manifest in reality.   I can hear you say again "but what do you mean by reality".  Well I must admit I haven't put a great deal of thought into this beyond what we say in the world of particle physics  where reality in effect is defined by the standard model (the zoo of matter particles, bozon's and whatever other exotica has recently been discovered).   I suppose we could distil that down a wee bit further to particular oscillations in fields at a quantum level and ultimately an expression of energy.  That is where, for me, we find reality.  I would also be happy to be told that these quantum fluctuations can be modelled mathematically using  in part the operator infinity;  I really would have no difficulty with that.  What I do have difficulty in accepting, and I have nothing other than a uneasy feeling to justify my position, is that infinity is not physically manifest in our reality.  We can model a bridge or a horse mathematically to our hearts content but it remains a mathematical model not a horse. 

I find infinity a fascinating concept, although the mathematics is beyond my level of understanding,  I'm  exploring well and truly down in the foothills.  I think my engineering background forever roots me firmly to the practical expression of nature so I am at a disadvantage here. For example, when we talk of relativistic speed,  the mathematical model will point that approaching the speed of light an object with non zero rest mass will increase in mass to infinity.  Similarly the energy taken to reach c would approach infinity. We are told that this clearly cannot happen - makes sense there cannot be a universe holding mass while my rocket ship has increased in mass to infinity. So we are told the equation is simply telling us we cannot  have objects with non zero mass accelerate to c.   My take on it is that the limiting factor is not our mass increasing to infinity rather the problem is the expression infinity itself imposing its conditions on reality.  Infinity needs to be contained in its  mathematical realm like a plasma in a magnetic bottle  

Jim  

 

 

  

 

   

 

 

[andrew s]

Oh course @saac you have at this very moment an infinity of masses depending on the velocity of the reference frame you measure it in with relative velocity v where |v| < c 😱

Regards Andrew