[Bill S (OP)]

Infinity is a mathematically conceptual number that can never be attained or written or imagined, except to say it is bigger than anything ever imagined. ie Big

A “conceptual number” does sound a bit like something that is not a number, but may be treated as though it were, if convenient; which is, sort of, what I have been saying about “mathematical infinity”.

[yor_on]

Einstein had this to say about 'infinity'

" If the universe is quasi-Euclidean, and its radius of curvature therefore infinite, then σ would vanish. But it is improbable that the mean density of matter in the universe is actually zero; this is our third argument against the assumption that the universe is quasi-Euclidean. Nor does it seem possible that our hypothetical pressure can vanish; the physical nature of this pressure can be appreciated only after we have a better theoretical knowledge of the electromagnetic field. According to the second of equations the radius, a, of the universe is determined in terms of the total mass, M, of matter, by the equation ***

The complete dependence of the geometrical upon the physical properties becomes clearly apparent by means of this equation. Thus we may present the following arguments against the conception of a space-infinite, and for the conception of a space bounded, universe


1. From the standpoint of the theory of relativity, the condition for a closed surface is very much simpler than the corresponding boundary condition at infinity of the quasi-Euclidean structure of the universe.


2. The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contained, to a first approximation, in the equations of the theory of relativity; it follows from these equations that inertia depends, at least in part, upon mutual actions between masses. As it is an unsatisfactory assumption to make that inertia depends in part upon mutual actions, and in part upon an independent property of space, Mach’s idea gains in probability. But this idea of Mach’s corresponds only to a finite universe, bounded in space, and not to a quasi-Euclidean, infinite universe. From the standpoint of epistemology it is more satisfying to have the mechanical properties of space completely determined by matter, and this is the case only in a space-bounded universe.

3. An infinite universe is possible only if the mean density of matter in the universe vanishes. Although such an assumption is logically possible, it is less probable than the assumption that there is a finite mean density of matter in the universe. The complete dependence of the geometrical upon the physical properties becomes clearly apparent by means of this equation. Thus we may present the following arguments against the conception of a space-infinite, and for the conception of a space bounded, universe  "

The 'pressure' discussed here comes from " Matter consists of electrically charged particles. On the basis of Maxwell’s theory these cannot be conceived of as electromagnetic fields free from singularities. In order to be consistent with the facts, it is necessary to introduce energy terms, not contained in Maxwell’s theory, so that the single electric particles may hold together in spite of the mutual repulsions between their elements, charged with electricity of one sign.

For the sake of consistency with this fact, Poincare´ has assumed a pressure to exist inside these particles which balances the electrostatic repulsion. It cannot, however, be asserted that this pressure vanishes outside the particles. We shall be consistent with this circumstance if, in our phenomenological presentation, we add a pressure term. This must not, however, be confused with a hydrodynamical pressure, as it serves only for the energetic presentation of the dynamical relations inside matter. "


The way I see it though it all seem to depends on what a 'universe' means.

https://www.gutenberg.org/files/36276/36276-pdf.pdf

[guest4091]

It's topics like this, where I recommend a dictionary as the primary reference.
Infinite translates to 'without end'.  It's an adjective (modifier), not a noun (thing).
Eg. The Peano axioms allow the formation of a larger integer, but not a largest integer.
Infinity is an attempt of the human mind to understand the meaning of something without end.
The problem is, the mind only has experience with finite things, with a beginning and an end.
The sequences of integers, 1 2 3 ...n, diverges, becomes larger than any preassigned n. 
Remember, all things imaginable, are not realizable.

[Bill S (OP)]

An infinite universe is possible only if the mean density of matter in the universe vanishes.

Why?  Simple explanation, please.

[chris]

Trying to explain something I think I understand to someone who thinks he doesn't, really gives the grey matter a good workout, and reading the opinions of experts in fields I haven't even thought of is just as much fun.

Hear Hear! 

[Bill S (OP)]

I'd really appreciate some guidance with #23.  I'm trying to tie up loose ends.

[alancalverd]

I think yor-on is confused!

A finite universe of massive objects would eventually coalesce to a single lump under their mutual gravitation, but an infinite universe would not, because each object sees an infinity of other objects: you might get local coalescence but the mean density would not change.

[alancalverd]

4. In calculus, a sequence like 1, 2, 3, 4 is considered as "converging to infinity", but is there actually an object "infinity" that this sequence is converging towards? 

Er, no. The term is "diverging". A convergent sequence has a limit: your sequence 1.1, 1.01, 1.001.... cannot have a term less than 1, and a series converging towards pi cannot sum to more than 3.142. But whatever number N you choose, the sequence 1,2,3,4....will eventually reach N + 1 and continue beyond it.

[andrew7278]

The process helped me a lot.

[andrew7278]

So, thank you very much.

[yor_on]

An infinite universe is possible only if the mean density of matter in the universe vanishes.

Why?  Simple explanation, please.

Bill, the way I read Einstein it would be that with a 'infinite universe' that term can't exist. A 'mean density' if I get it right presumes a border from where you can measure said 'density' inside it. Maybe there is a better explanation but that's the way I read it then. And no Alan, if there was anyone confused you better take it up with Einstein. Which might be slightly difficult at present.

[yor_on]

Then again, possibly he was thinking of something different? Because we can imagine a infinity populated by mass, and make models in where this 'mean density' will differ, as long as we presume the mass to be evenly distributed everywhere, aka a 'isotropic and homogeneous universe'. We don't need borders for that one, do we?
=

Or maybe we actually need it?
I'm not sure?

[Bill S (OP)]

Er, no. The term is "diverging". A convergent sequence has a limit: your sequence 1.1, 1.01, 1.001.... cannot have a term less than 1

Trying to do too many things at once – didn’t really say what I meant.
I’ll try again.  The sequence, 1.1, 1.01, 1.001.... converges to 1, so has a finite limit. However, the number of “steps” has no limit; I can always add another “0”; so the number of “steps” may be considered as infinite.

But whatever number N you choose, the sequence 1,2,3,4....will eventually reach N + 1 and continue beyond it.

Isn’t that tantamount to saying that the sequence is “infinite”?  If so, that seems to accord with the point I was trying to make; albeit falteringly.

[alancalverd]

So you have at least two definitions of infinity

(a) the number of iterations required to make an infinite series converge to its limit

(b) a number larger than any other number

and because one is convergent and the other divergent, they are different sizes!

[Bill S (OP)]

So you have at least two definitions of infinity

Exactly!

Liking simplicity, I am inclined to think in terms of two basic interpretations of infinity.  I see no problem with either, as long as no one confuses the two. 

Neither is a number, but one includes concepts such as infinite sets, infinite sequences etc.  The other is often obscured by philosophical, and even religious overtones, but is essentially physical, in that it answers the very basic question: How can there be something, rather than nothing?

(a) the number of iterations required to make an infinite series converge to its limit

If a series is infinite, isn’t its “limit” unattainable?

(b) a number larger than any other number

Isn’t this saying that infinity is a number?

[alancalverd]

(a) the number of iterations required to make an infinite series converge to its limit

If a series is infinite, isn’t its “limit” unattainable?

Yes. But definable! (NB we must distinguish between a convergent sequence, which converges towards a terminal value, and a convergent series whose sum converges towards a terminal value. 1.1, 1.01, 1.001.... converges towards 1, but 1 + 1.01 + 1.001... has no limit, whilst 1 + 1/2 + 1/3 +1/4....converges to e)

(b) a number larger than any other number

Isn’t this saying that infinity is a number?

There are all sorts of infinities, most of which have the genesis of a number.

I think Cantor implied, even if he didn't prove, that there is an infinite number of infinities.

You can start with arithmetic, by listing integers 0,1,2, 3.... and for any integer N  you can propose an integer N+1, so that's your first infinity - the number of possible integers, ℵ₀ .

Now consider the interval between 0 and 1. You can fill it with rational numbers 1/N, 1/(N-1), 1/(N-2)…. and we have already shown that there is no limit to N, so there are ℵ₀  rationals between any two integers, and the number of possible rationals is therefore a larger infinity, ℵ₁.

But between any two rationals, you can insert any number of irrationals, giving you a yet larger infinity ℵ₂…

and so, ad infinitum. Now each point within these infinities has a numeric value, so they are "denumerable infinities". However finely you divide the intervals, you can cut loose from arithmetic and conceive of a continuum that incorporates all the divisions plus an infinitesimal gradation between them, leading to a nondenumerable infinity - the province of differential calculus and classical physics, wherein the delta function, Fourier transform, and a whole lot of other useful stuff, resides.

Cantor went mad. Don't follow him. But do use differential calculus!

[Bill S (OP)]

we must distinguish between a convergent sequence, which converges towards a terminal value, and a convergent series whose sum converges towards a terminal value. 1.1, 1.01, 1.001.... converges towards 1, but 1 + 1.01 + 1.001... has no limit, whilst 1 + 1/2 + 1/3 +1/4....converges to e

Point taken, but differentiating between a sequence, which converges towards a terminal value that it can never reach, one that converges towards no identifiable limit and one that converges towards an endless decimal fraction, does seem to be making little more than a conceptual distinction, however valuable the concepts might be in mathematics.

 

There are all sorts of infinities, most of which have the genesis of a number.

 “You can’t lie with math. But it greatly aids obfuscation.”  Sabine Hossenfelder   

I think Cantor implied, even if he didn't prove, that there is an infinite number of infinities.

That makes perfect sense to me. Perhaps, if I were a mathematician, my thought patterns would be similar to Cantor’s.  It was his concept of “Absolute infinity” that did much to help me to reconcile the two main “definitions” of infinity.  BTW, how would you prove that there was an infinite number of anything?

Cantor went mad. Don't follow him. But do use differential calculus!

I’m afraid it’s too late, both for the sanity and for learning calculus. About three years ago I downloaded a calculus course. I’ve still not found the time to start it.

[alancalverd]

differentiating between a sequence, which converges towards a terminal value that it can never reach, one that converges towards no identifiable limit and one that converges towards an endless decimal fraction, does seem to be making little more than a conceptual distinction, however valuable the concepts might be in mathematics.

It's enormously important!

In the first case I can write down a rational number (in this case 1) that is just a bit larger or in this case smaller than  any term in the sequence. So if you reported your position every hour as 1.1, 1.01, 1.001...miles from my house, I would know exactly where to look for you.

In the second case, I would know you are horribly lost and not recoverable because you are moving away from me at about 1 mph.

In the third case, although I couldn't write down a rational number that tells me where you would end up, I know it would be less than 2.7183 miles away.

[Bill S (OP)]

Thanks Alan.  I suspected it would be mathematically significant, and as Maths is the best description we have of our Universe, it would also be physically important.  It’s good to have some examples to confirm the suspicions, though.   

[Bill S (OP)]

I’ve put together a few questions, and made some tentative moves towards possible answers, or where they might be found.  Still a long way to go, but comments would be appreciated.
 
1. If infinity is not a number, how can you subtract anything from it?

Only mathematical infinities are amenable to mathematical processes.

There’s an interesting philosophical comment here:

https://philosophy.stackexchange.com/questions/29220/how-can-numbers-be-infinite

2. If the Universe is infinite, it contains an infinite number of galaxies.  How does one define an infinite number?

There is no such thing as an infinite number.  The term is “shorthand” for: There is no limit to the number you could, potentially, observe, if you could continue observing for ever.  As no one has continued observing/counting “for ever”, nor is anyone likely to, the “infinite number” is conceptual.  However, the fact that I continue to use the term demonstrates its practical usefulness.

3. If one is subtracted from an infinite number of objects, is the remainder still an infinite number?  If not, what is it?

https://www.quora.com/Why-does-1-subtracted-from-infinity-equal-infinity

You want to show people just how interesting mathematics is, but at the same time you cannot avoid fudging some details and blurring the truth if you want to do it in a 2 minute clip or half-page article in some magazine.

4. Would an infinite number of (identical) objects contain all the examples of that object that could exist?

As infinity is not a number, the question is meaningful only within the “mathematical definition” of infinity; in which sense, the answer must be “yes”.

5 Is “absolute infinity” (sensu, Cantor) amenable to mathematical manipulation?

No.  If it were, it would not be “absolute infinity”.

6.

The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor.
It can be thought as a number which is bigger than any conceivable or inconceivable quantity, either finite or transfinite.

How could this concept be expressed without referring to infinity as “a number”?

I’m going to leave that to those with greater knowledge than I have.