Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Bill S on 10/03/2019 12:25:20
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There’s a chance I may not continue posting, and there are a few things I would still like to clarify, just in case. Please have patience if I dig up some “oldies”.
Looking back at past exchanges, I find myself thinking that there are areas in which we tend to go round in circles. This may be because some of us “hitch-hikers” are slow to catch on. It may also be because experts often answer the question they think should have been asked, rather than the one that was actually asked. Then there are crossed wires……
One point that needs to be cleared up arises from: Can we lay nothing to rest? https://www.thenakedscientists.com/forum/index.php?topic=53002.msg445560#msg445560 Where Pete says:
But you haven't been appreciative enough to take my advice and read what I suggested. I feel as if you're being disrespectful to me when the fact is that I'm being as respectful to you as I possibly can by doing a lot of work trying to help you understand this and all you do is skirt around my one single suggestion which will help you
It saddens me that, in asking a question about something that an expert has said, I might give the impression that I am disrespecting that person’s expertise. Such is not the case, but I will never pretend that I understand something, if I don’t. Therefore, I ask questions.
I did read the information Pete attached, it certainly improved my understanding of mathematical infinities; thanks Pete. However, there still remain a few unanswered questions. This may be, in part because, before we can say anything meaningful about a concept, we have to be quite clear about the context in which a question is being asked, and an answer given.
Before even attempting to go any further, I should check, and possibly improve, my understanding of the mathematical concept of infinity. Hopefully I will post my thoughts soon.
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Before even attempting to go any further, I should check, and possibly improve, my understanding of the mathematical concept of infinity. Hopefully I will post my thoughts soon.
That path leads to insanity, as many mathematicians have shown!
However most physicists and engineers seem to die with their brains relatively intact, having decided that infinity is a tool determined by context and either means "Limitx→0 {A/x}" where A is pretty well anything (physics) or "too big or far away to make any practical difference" (engineering).
Your departure would diminish the challenge and value of these boards! 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.
Just as Eddington said "the student of physics must become accustomed to having his common sense violated five tims before breakfast", so anyone paddling in the shoals of science must be prepared for some fairly robust exchanges with the whales that explore the deep waters.
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You are one of our valued members Bill. Don't ever forget that. In amongst all the madness that ensues you are a voice of reason. We have too few as it is.
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Alan and Jeffrey, thanks for the encouraging comments. There are times when I feel it would be less frustrating, just to draw a line under all this, rather than keep “picking” at it; but I’m not really that sort of person, so I’ll try to keep a toe in the door.
Insanity is not a problem for me – I know I’m mad; so here are some thoughts about mathematical infinity, inviting criticism.
1. Mathematical objects do not exist in the same sense that physical objects exist. “If a mathematician could write down a set of non-contradictory axioms and rules for deducing true statements from them, then those statements would be said to ‘exist’.” (Barrow)
2. Infinity is not a number. It may be found used in such a way that it looks like it is being treated as a number, but this probably arises because of “shorthand” terminology. Eg. Process “A” is effectively endless, so it is referred to as “infinite”. This does not claim that there is a point, identified as “infinity” that can actually be reached. This would be a contradiction in terms.
3. Infinity is meaningful in terms of a topological space in which two sequences of objects converge, but never meet. Thus, the sequence 1.1, 1.01, 1.001, 1.0001, etc. converges to the number 1, but never reaches 1. The “shorthand” way of expressing this is to say that the sequence is infinite.
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?
Mathematically, there is. If you can define a topological space which has the real numbers as a subset, plus an additional object called "infinity", and if the notion of convergence is such that sequences converge to this additional object. In that sense, infinity exists. (I don't believe I said that :) )
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In that sense, infinity exists. (I don't believe I said that )
Does "infinity" look very small when viewed from afar (eg written as a symbol) but balloon to the size of the whole universe when approached in detail?
A bit like my seemingly wrong idea of "Russian-dolled" events in the "events" thread I just started.... where the fall of Rome seems like one event from a distance .
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Infinity exists in the same way that God exists: it is a word that we use as we wish, to convey whatever is appropriate in context.
E.g. "God knows" = there may be a rational explanation for the status quo but looking for it won't help us solve the present problem. "Focus to infinity" = set up to converge parallel beams to the focal point.
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Yes, it seems pedantic to have to repeat that infinity is a mathematical process rather than a thing in its own right (or a word that can be used loosely to convey various useful ideas)
But it seems that there may be a natural inclination to want to make it into or picture it as a real thing when it is not.
Even so,it does seem a genuine question whether or not the universe itself is finite or infinite (I go for infinite-and am also happy with the "infinite" level of lack of understanding that quite probably entails)
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Infinity exists in the same way that God exists: it is a word that we use as we wish, to convey whatever is appropriate in context.
So, where does that leave my attempt to find a description of mathematical infinity?
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Infinity exists in the same way that God exists: it is a word that we use as we wish, to convey whatever is appropriate in context.
“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”
The original Humpty Dumpty may have been a cannon, but, could be, this one was a scientist. :)
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So, where does that leave my attempt to find a description of mathematical infinity?
Try Brian Clegg's "Brief History of Infinity" - a very readable summary of most infinities, including the reason why your search is doomed!
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I should probably re-read that, but at this point, I was just considering my understanding, or otherwise, of the concept of mathematical infinity. One step at a time.
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I learned that one mathematical infinity can be bigger than another. It was schoolboy stuff ,if a little above my head then.
Are there any practical applications of this finding or is it purely theoretical? (even if obviously true)
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If you can define a topological space which has the real numbers as a subset, plus an additional object called "infinity", and if the notion of convergence is such that sequences converge to this additional object. In that sense, infinity exists. (I don't believe I said that :) )
Infinity is number that can never be attained, a distance that can never be travelled in an eternity, a time that can never be reached. ie Infinity has no limit it is unending.
Ps it is also useful in maths to simplify equations as f(a value) → ∞ something can be ignored in an equation, making the sums easier.
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Infinity is number that can never be attained,
Yes, we do go round in circles. Infinity, it seems, is not a number, unless we choose to treat it as a number.
As Alan pointed out: "....it is a word that we use as we wish, to convey whatever is appropriate in context.".
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Infinity is never a number. If there is ever only one thing you learn about infinity that should be it. What is 5 times infinity? It is undefined in mathematical terms.
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You are preaching to the converted, there, Jeffrey.
2. Infinity is not a number.
Just one of the many times I've said it; usually in response to its being used as though it were a number.
BTW; if you have any comments about the points in #3, I would welcome them.
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You are preaching to the converted, there, Jeffrey.
2. Infinity is not a number.
Just one of the many times I've said it; usually in response to its being used as though it were a number.
BTW; if you have any comments about the points in #3, I would welcome them.
OK
Infinity is concept of a number that can never be attained ie N/0 →∞. Your number sequence given in 3 is correctly infinite although the number 1.01, 1.001 ultimately → 1.0
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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? " It is converging towards the concept of a number that can never be attained.
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
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Infinity is never a number. If there is ever only one thing you learn about infinity that should be it. What is 5 times infinity? It is undefined in mathematical terms.
Maybe my mind is slipping but "undefined" would be something that is undefined in mathematics, ie 0/0 0^0 can not be defined etc N/0 →∞ is not strictly undefined, in that we know which way the number is headed, if we were to try and calculate the outcome. Although ∞/∞ would be undefined like 0/0 https://en.wikipedia.org/wiki/Undefined_(mathematics)
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My understanding is that “undefined”, like other terms we have been looking at, is used in a variety of ways. This can be very convenient, and facilitate scientific progress, or it can be bl**dy frustrating, depending on your perspective.
We have been considering the nature of a point, and its connection with an event. It surprises me that no one has “slammed the door” on this by asserting that the term “point” is undefined.
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My understanding is that “undefined”, like other terms we have been looking at, is used in a variety of ways. This can be very convenient, and facilitate scientific progress, or it can be bl**dy frustrating, depending on your perspective.
We have been considering the nature of a point, and its connection with an event. It surprises me that no one has “slammed the door” on this by asserting that the term “point” is undefined.
I thought you had defined the point of the OP :)
The point is "a single item or detail in an extended discussion, list, or text" An event is something that occurs at a point in space time. :)
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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”.
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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
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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.
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An infinite universe is possible only if the mean density of matter in the universe vanishes.
Why? Simple explanation, please.
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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! :)
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I'd really appreciate some guidance with #23. I'm trying to tie up loose ends.
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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.
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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.
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The process helped me a lot.
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So, thank you very much.
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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.
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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?
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Or maybe we actually need it?
I'm not sure?
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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.
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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!
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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?
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(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!
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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.
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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.
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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.
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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.
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A infinity shouldn't have borders. But then we have mathematical infinities in where we can define one as larger than the other.
Two choices.
1. it's correct
2. it's not
It makes one wonder Bill. Assume that it is correct. Then ask yourself of what magnitude out presumably 'infinite universe' is? And how you would build one.
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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.
I put in some comments, but have not read the whole thread.
1. If infinity is not a number, how can you subtract anything from it?
It isn't meaningful to subtract a number from a not-a-number. 'Infinity' is more of an adjective, meaning 'without limit'. Much confusion arises when its syntactic usage as a noun leads one to treat it as a number.
2. If the Universe is infinite, it contains an infinite number of galaxies. How does one define an infinite number?
Again, it simply means there is no limit to the number of galaxies.
3. If one is subtracted from an infinite number of objects, is the remainder still an infinite number? If not, what is it?
Per point 1, it is not meaningful to do addition and subtraction with 'without limit'.
4. Would an infinite number of (identical) objects contain all the examples of that object that could exist?
If they're identical, how is it not one object? This sort of gets into the law of identity.
Perhaps you mean something like 'just because there are infinite points along a line doesn't mean that there are not other points that do not fall on that line. So there are examples of points not in that set, but I'd not call any of the points 'identical' since they're all at different places on the line.
5 Is “absolute infinity” (sensu, Cantor) amenable to mathematical manipulation?
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”?
Since it isn't a number, not sure what you're asking. It's expressed as Ω, but that's not your question. One infinity isn't larger than another since they're not numbers. Perhaps it means 'highest cardinality', but cardinality isn't an expression of the magnitude of something. Only numbers have magnitude.
I'd not trust wiki on this. They call it a number, but any number is finite. I think a proper mathematician would not word the description this way.
An infinite universe is possible only if the mean density of matter in the universe vanishes.
Why? Simple explanation, please.
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I'd really appreciate some guidance with #23. I'm trying to tie up loose ends.
Since nobody replied to this, I'll just say that the mean density of matter in the universe is typically presumed to be the same as it is in the parts we see (cosmological principle). yor-on's statement would only apply to a model where there is infinite space, but there is finite matter, presumably all clumped nearby. I know of no such model that is seriously considered.
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Halc, I’ve just managed to read your post and compare your comments with mine. Allowing for differences of interpretation and understanding, there seemed to be a reasonable concordance. A couple of points need clarification.
Would an infinite number of (identical) objects contain all the examples of that object that could exist?
If they're identical, how is it not one object?
How do you define “identical”? My understanding would be that it means “exactly alike”. How could one object be “exactly alike”?
Perhaps you mean something like 'just because there are infinite points along a line doesn't mean that there are not other points that do not fall on that line.
No, that’s not what I meant. I consider that to be a valid statement; it involves mathematical infinities, of which there are more than enough to make that work.
I was trying to achieve some clarity regarding the definitions of infinity.
If infinity is not a number, how can you have an infinite number of anything?
In maths, you can talk, legitimately, of “infinite points along a line”, this is treating infinity as a number, but I accept that as a valid mathematical device. However, if we talk of an infinite universe, and we acknowledge that that universe contains objects, we must be saying that there are “infinite” objects.
The term, “infinite objects” differs from “an infinite number of objects” only semantically.
I’ll rephrase my question. If the Universe is infinite, and contains (an) infinite (number of) galaxies; could there be other galaxies, that are not included in this?
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I’ll rephrase my question. If the Universe is infinite, and contains (an) infinite (number of) galaxies; could there be other galaxies, that are not included in this?
If they are not in the set of galaxies in our universe then where are they?
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How do you define “identical”? My understanding would be that it means “exactly alike”. How could one object be “exactly alike”?
As I said, it depends on your definition of identical. Two objects in different places have different coordinates, and thus not 'exactly alike', however much you'd not be able to tell them apart if they were scrambled around. If they have the same coordinates, they're the same object, no?
Would an infinite number of (identical) objects contain all the examples of that object that could exist?
Perhaps you mean something like 'just because there are infinite points along a line doesn't mean that there are not other points that do not fall on that line.
No, that’s not what I meant. I consider that to be a valid statement; it involves mathematical infinities, of which there are more than enough to make that work.
I thought I gave a counter example. The set of points on a mathematical line is not a finite set, and yet there are points not on that line, which are examples of a points that can exist that are not in the first (infinite) set. So the first infinite set does not contain all examples of that object (points) that can exist since I was able to find object not in the set. 'Infinite' does not imply 'all'.
I was trying to achieve some clarity regarding the definitions of infinity.
If infinity is not a number, how can you have an infinite number of anything?
You can't. There is not a number that represents the total count of integers for instance.
In maths, you can talk, legitimately, of “infinite points along a line”, this is treating infinity as a number but I accept that as a valid mathematical device.
You're doing your maths wrong if you treat that as a number. I do not accept it as a valid mathematical device.
However, if we talk of an infinite universe, and we acknowledge that that universe contains objects, we must be saying that there are “infinite” objects.
The term, “infinite objects” differs from “an infinite number of objects” only semantically.
The latter form is invalid. Saying 'there are infinite objects' just says that there is no number that represents the count of these objects. I know people us the phrase 'infinite number' all the time in casual conversation. We know what they mean. But in mathematics, that is invalid.
I don't hold a mathematics degree (despite taking 3rd place in a high-school mathematics competition for a region of about 10 million people)
I’ll rephrase my question. If the Universe is infinite, and contains (an) infinite (number of) galaxies; could there be other galaxies, that are not included in this?
Get rid 'number of' to let it make syntactic sense.
Yes, there could be galaxies not included in 'the universe'. Maybe there are other universes. I personally don't buy into the existence of unmeasured galaxies, and hence I would say there can only be a finite number of them, but that's just me and my personal definition of 'existence' and 'universe' and such. All these words have different meaning to different people.
So I talk about integers when I need an example of something infinite, and there are indeed numbers that are not included in that infinite set. Those numbers are no more or less identical than 5 is from 13.
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If the Universe is infinite, and contains (an) infinite (number of) galaxies; could there be other galaxies, that are not included in this?
That depends on your definition of universe. To most people, it means "everything", or, if you like, "every thing", so there can't be any things that are not members of the universe.
If you are just counting galaxies as "things", then an infinite universe consists of a denumberable infinity of galaxies, which as Cantor pointed out, is one of the smaller infinities - possibly the smallest, but now I'm scraping the barrel of my memory!
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If (two objects) have the same coordinates, they're the same object, no?
In the context of this thread (counting massive galaxies), that is correct, as is its converse.
But in the spooky quantum world of microscopic objects, the converse is not necessarily so.
- In the dual-slit experiment, we have two slits at different coordinates in space
- But the wavicle which passes through these slits is (in some sense) at both coordinates, before it is detected
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If (two objects) have the same coordinates, they're the same object, no?
In the context of this thread (counting massive galaxies), that is correct, as is its converse.
But in the spooky quantum world of microscopic objects, the converse is not necessarily so.
- In the dual-slit experiment, we have two slits at different coordinates in space
- But the wavicle which passes through these slits is (in some sense) at both coordinates, before it is detected
There are two slits, but I'd not consider them identical since one is a left slit and the other a right slit.
As for the wavicle (is that a word? I like it), most interpretations say the light is in superposition of passing through both slits, which is not the same as the counterfactual statement of it actually passing through any particular slit. An interpretation that makes such counterfactual assertions (like pilot wave theory) says the wave goes through both slits (just like water would), but the particle goes through one of them and then rides the wave pattern on the other side. All interpretations making such statements also need to posit action at a distance (non-locality).
I'm more in the non-spooky camp that says no action at a distance or alteration of the past, and hence I cannot make such counterfactual statements. But again, that's just my opinion, not a demonstrable thing.
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That depends on your definition of universe. To most people, it means "everything", or, if you like, "every thing", so there can't be any things that are not members of the universe.
I agree, but if you differentiate between the Universe and the cosmos (sensu Gribbin); there can be things that are not part of the Universe, but must be part of the cosmos. In some ways, that complicates things, but in others, is introduces simplicity.
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I haven't pursued Gribbin (never liked his early books) but Wikipedia is succinct
Using the word cosmos rather than the word universe implies viewing the universe as a complex and orderly system or entity; the opposite of chaos.
I think this stinks of philosophical arrogance and mathematical ignorance!
Everything affects everything else to some extent. We can (a) predict the outcome of an idealised 2-body interaction with plenty of confidence, and also state (b) that for any finite sample of the universe, entropy increases with time. The equation of (b) with chaos, or even the suggestion that the universe is or is increasingly chaotic, is simple defeatism.
We provably cannot predict all the interactions of all the particles in the universe, but we have no reason to think that any one interaction is not subject to the same laws (a) as all the others. Chaos is therefore in the eye of the beholder.
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Well, a ordered shelf with the socks neatly stacked in pairs is of a low entropy if I remember right. With time those socks will get lost, unordered or otherwise disappearing from the shelf, that's a higher entropy. We have one arrow, it always point in a 'one way direction' and with it we will meet a raising entropy. Using entropy you might define it as going towards a place of a 'higher' unordered entropy, aka 'chaos'. But that's only half a truth, we can have places of low entropy inside a system of high entropy and the end result, of a system of high entropy, becomes pretty 'ordered' to me. It's like the 'heat death' of the universe in where everything is equivalent, nothing more happening. So entropy actually hurts my head.
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I haven't pursued Gribbin (never liked his early books)
My first Pop Sci book was his “In Search of Schrödinger’s Cat”. I thought it was a bit like the “curate’s egg”. I tried a couple of others and liked them even less. However, I think his distinction between Universe, universe and cosmos can be quite helpful; which is, probably, more than be said of the Wiki quote.
Making the distinction between the (apparently) finite Universe and an infinite cosmos can, with some unbiased thought, help to avoid at least three common pitfalls, when thinking about how there can be “something”, now.
1. Treating nothing as though it were something.
2. Treating infinity as though it were a number, while strenuously insisting that such is not the case.
3. Resorting to thaumatology when all else fails.
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Like the frequently encountered deck of cards analogy, this can easily give a wrong impression. Just changing the position or orientation of macroscopic objects from an arrangement that we define as orderly, to one that we define as disorderly doesn’t necessarily change its entropic state. We have to ask ourselves: does moving a sock alter the number of microstates accessible to its constituent particles? Does being shuffled, or unshuffled, change the microstates of a deck of cards?
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It's all a matter of probabilities.
If you define order in terms of adjacency, then pairs of socks are first-degree ordered: any red sock is more likely (indeed certain) to be adjacent to another red sock than any other color. If you now place the pairs in a rainbow, you have a second-degree order because if you start with a red sock you know exactly where to look for a yellow sock, and indeed any other color. The definition of a random set is that knowledge of A gives you no clue as to the nature of its neighbour B or of any other member of the set, unless the set is bounded (published "random number tables" are limited to 100 or 1000 numbers with an even but wholly unpredictable distribution)
So moving a sock doesn't alter the entropy of the set if the move was deliberate and documented, but shaking the box transforms the distribution from a highly improbable spectral order to a mess which is just as probable as any other mess after the same amount of shaking.
Now here's a fun thought, just as I rush out of the door. We can enumerate all the throws of two dice, and easily conclude that the most probable score is 7. What is the most probable score for 3 dice? for n dice?
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So moving a sock doesn't alter the entropy of the set if the move was deliberate and documented, but shaking the box transforms the distribution from a highly improbable spectral order to a mess which is just as probable as any other mess after the same amount of shaking.
I can see the analogy involving the distribution of socks, and the fact that ordered movement and shaking can produce very different outcomes, but, surely, even shaking does no more than change the relative positions of macroscopic objects. Both processes involve an increase in entropy, but that increase is a property of the agent of change, rather than the objects themselves.
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Nice question Alan. But there is this thing with probabilities, how many throws will you need to get the right statistics?
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Both processes involve an increase in entropy
No. Intentionally moving one sock to as known position results in a new but entirely predictable order, hence no increase in entropy. It is just as improbable as the original order!
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Nice question Alan. But there is this thing with probabilities, how many throws will you need to get the right statistics?
We're back to converging sequences! The more throws, the greater your confidence that your measured distribution corresponds to the "true" distribution for an infinite number of throws.
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On chaos, if an irrational number has a constant, but unpredictable, sequence of digits, then it cannot be said to be chaotic. If the universe is deterministic, it also cannot be said to be chaotic. So what part does indeterminacy play in whether or not the universe is chaotic?
EDIT: Is indeterminacy in the eye of the beholder only?
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I tend to reserve indeterminacy for the properties of quantum systems, in particular electrons. It is an inherent property of the system, not the observer. If indeterminacy depended on observation, hydrogen atoms would collapse to neutrons and organic molecules would collapse to a spherical blob when nobody is looking at them, then miraculously leap back into shape when they discover that you care. Guillaume d'Ochambe would be disgusted.
The universe is clearly ordered locally, to the extent that we can predict idealised two-body interactions to any degree of accuracy allowed by essential quantum indeterminacy, but the fact that we can't know about a cause until we see the effect means that it is macroscopically chaotic: (a) you don't know about the butterfly until the hurricane hits you, and (b) unravelling the chain of events is unlikely to lead you back to the butterfly.
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Both processes involve an increase in entropy…..
No. Intentionally moving one sock to as known position results in a new but entirely predictable order, hence no increase in entropy. It is just as improbable as the original order!
….. but that increase is a property of the agent of change, rather than the objects themselves.
Could we be saying the same thing?
Changing the arrangement does nothing to the entropy of the socks, but the person making the changes increases global entropy.
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Spontaneous increases in entropy only occur when a system moves to a lower energy state. Doing work by moving a sock certainly increases the entropy of the universe.
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We have to ask ourselves: does turning a coin over alter the number of microstates accessible to its constituent particles? Does being shuffled, or unshuffled, change the microstates of a deck of cards?
The answer, in both cases, has to be, no.
OK, I will climb off this particular soap box now, and in so doing, will slightly increase the entropy of the Universe. :)