Is infinity a misconception?

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Offline jeffreyH

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Is infinity a misconception?
« on: 17/09/2014 22:19:51 »
If we have a right angled triangle and one side adjacent to the right angle is finite while the other adjacent side is infinite then the hypotenuse must be greater than infinity. What does this say about our view of infinity?

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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #1 on: 17/09/2014 22:41:42 »
It says nothing about "our" view of infininty, but speaks volumes about your understanding of the principal axioms of Euclidean geometry.  The key construction for analysing this triangle is of course to draw a circular square centered on the bisector of the infinite side, using the usual methods.
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Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #2 on: 17/09/2014 22:59:07 »
Well then find the radius of the circle whose circumference is infinite.

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Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #3 on: 17/09/2014 23:04:19 »
Conversely if the radius is infinite find the circumference.

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Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #4 on: 17/09/2014 23:34:17 »
The point here is that any system that can normally be considered as bounded cannot include an infinite component. So if the mathematics of a formerly finite system go infinite somethings is terribly wrong. If we consider the event horizon as our boundary then the black hole can be thought of like a superconductor.

http://arxiv.org/abs/1403.0938

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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #5 on: 18/09/2014 01:54:17 »
Quote from: jeffreyH
If we have a right angled triangle and one side adjacent to the right angle is finite while the other adjacent side is infinite then the hypotenuse must be greater than infinity. What does this say about our view of infinity?
Nothing. I think that you're confusing infinity with a number. It means increases without bound. Let me make this very clear first; [tex]\infty[/tex] is not a number.

It's defined as follows;
Quote
The notation

[tex]\lim_{x \to \infty} f(x) = 0[/tex]

means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side) but not equal to a.

Most of the time [tex]\infty[/tex] is used without this detail and in a more at ease manner (i.e. sloppy). When you start studying calculus and you get into limits you'll have to evaluate expressions like f(x)/g(x) where both f(x) and g(x) increase without bound. Sometimes the ratio approaches a limit and sometimes it doesn't. If you choose to study calculus you'll learn L'Hôpital's rule which is used to handle these cases.

http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule

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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #6 on: 18/09/2014 06:36:43 »
The point here is that any system that can normally be considered as bounded cannot include an infinite component.

Precisely. A triangle is by definition bounded by three intersecting sides, so your "triangle with one infinite side" is as meaningless as my "circular square centered on the bisector of the infinte side".

Just because words hold their conventional order in a phrase doesn't imply that the phrase means anything.   

« Last Edit: 18/09/2014 06:42:43 by alancalverd »
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Offline JohnDuffield

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Re: Is infinity a misconception?
« Reply #7 on: 18/09/2014 12:55:40 »
If we have a right angled triangle and one side adjacent to the right angle is finite while the other adjacent side is infinite then the hypotenuse must be greater than infinity. What does this say about our view of infinity?
I'm not sure that's a great example, but I agree with the thrust of what you're saying. I think infinity is a misconception. IMHO if ever you bump into an infinity in physics, then something is wrong somewhere. As a rule of thumb, there are no infinities in nature. For example, gravitational time dilation is said to be infinite at the black hole event horizon. But did you see the expression pmb referred to?

[tex]\lim_{x \to \infty} f(x) = 0[/tex]

IMHO it's better to say that the coordinate speed of light is zero at the event horizon.
« Last Edit: 18/09/2014 12:57:22 by JohnDuffield »

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #8 on: 18/09/2014 17:32:30 »
Quote from: Pete
I think that you're confusing infinity with a number. It means increases without bound. Let me make this very clear first;  is not a number.

“Infinity is not a number”!  It’s good to hear someone else make that assertion.

“It means increases without bound”.       Would it not be better to use “boundless”?   

Infinite and boundless are not synonymous.

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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #9 on: 18/09/2014 18:22:20 »
Here's an old but very strong example (Cantor?)

There is an infinite number of integers 1,2,3,.... because we can always add one more

There are rational numbers between the integers 1, 3/2, 7/4, 2, 9/4, 19/8, 3....

Indeed there is an infinite number of rational numbers between any two integers

So the number of rational numbers must be greater than the number of integers

So there are at least two classes of denumerable infinity, even in one dimension

And we can fill the spaces between rational numbers with nonrational numbers.....
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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #10 on: 18/09/2014 18:37:54 »
Quote from: Bill S
“It means increases without bound”.       Would it not be better to use “boundless”?   
Why do you think it'd be better?

Note: The infinity sign is missing when you quoted me. Instead of reading
Quote
Let me make this very clear first;[tex]\infty[/tex]is not a number.
Your quote instead reads
Quote
Let me make this very clear first;  is not a number.
« Last Edit: 18/09/2014 18:41:18 by PmbPhy »

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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #11 on: 20/09/2014 03:26:58 »
Quote from: JohnDuffield
IMHO if ever you bump into an infinity in physics, then something is wrong somewhere.
Nonsense. It's beginning to become clear that the universe is flat and boundless and as such goes on forever, never ending. That's what it means to be infinite. It also appears to have approximate uniform mass density which means that there's an infinite amount of hadrons in the universe too.

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Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #12 on: 20/09/2014 11:34:32 »
Quote from: JohnDuffield
IMHO if ever you bump into an infinity in physics, then something is wrong somewhere.
Nonsense. It's beginning to become clear that the universe is flat and boundless and as such goes on forever, never ending. That's what it means to be infinite. It also appears to have approximate uniform mass density which means that there's an infinite amount of hadrons in the universe too.

In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. So to all intents and purposes the contents of the universe is finite even though the universe itself may not be. Given an infinite amount of time there is the possibility of every particle in the universe interacting with every other particle, every field interacting with every other field. The same exact combination of particles also have an infinite time in order to interact in exactly the same way more than once. I am not disagreeing with you here Pete I think this is a fascinating subject.

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Offline JohnDuffield

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Re: Is infinity a misconception?
« Reply #13 on: 20/09/2014 15:09:50 »
Nonsense. It's beginning to become clear that the universe is flat and boundless and as such goes on forever, never ending. That's what it means to be infinite. It also appears to have approximate uniform mass density which means that there's an infinite amount of hadrons in the universe too.
It's beginning to become clear that the universe is flat, but it absolutely isn't clear that it goes on forever. IMHO this is a non-sequitur promoted by cosmologists who have an inadequate understanding of general relativity.

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #14 on: 20/09/2014 19:21:10 »
Pete, sometimes you surprise me.

“Without bound” and “boundless” are synonymous.  I doubt that anyone would argue with that.

“Without bound” and “infinite" may mean the same in certain cases, but there would be at least one person on this forum who would argue with any claim that they are synonymous.

Why introduce doubt unnecessarily?

You are absolutely right about the missing infinity sign, which is odd, as I cut and pasted the quote.  I apologise if this caused you disquiet; but I was agreeing with you. 

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #15 on: 20/09/2014 19:37:02 »
Quote from: alancalvard
Here's an old but very strong example (Cantor?)

There is an infinite number of integers 1,2,3,.... because we can always add one more

The sequence of integers is boundless…. “because we can always add one more”.
It is quite reasonable to refer to this as “infinite”, as long as everyone recognises that this is a mathematical infinity – not some sort of physical infinity.

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #16 on: 20/09/2014 19:44:41 »
Quote from: JD
It's beginning to become clear that the universe is flat, but it absolutely isn't clear that it goes on forever. IMHO this is a non-sequitur promoted by cosmologists who have an inadequate understanding of general relativity.

I believe it also arises from unclear thinking about the way in which a physical infinity (if it exists) would differ from a mathematical infinity; which, as Cantor demonstrates, does “exist”.   

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #17 on: 20/09/2014 19:50:35 »
Quote from: alancalvard
Just because words hold their conventional order in a phrase doesn't imply that the phrase means anything.

That I like!  I hope you will not mind if I borrow it at some time.

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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #18 on: 20/09/2014 23:41:02 »
It is quite reasonable to refer to this as “infinite”, as long as everyone recognises that this is a mathematical infinity – not some sort of physical infinity.


Here's a physical infinity. The gravitational force F exerted by an object of mass m decreases as m/r2, so F→0 as r→∞ . Physical reality? Well we can measure any  F > 0, so it's real 

Now the force exerted by a mass 2m decreases as 2m/r2, so F→0 as r→∞' and clearly ∞' > ∞
« Last Edit: 20/09/2014 23:47:08 by alancalverd »
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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #19 on: 21/09/2014 01:35:20 »
Quote from: jeffreyH
In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. So to all intents and purposes the contents of the universe is finite even though the universe itself may not be.
You're reasoning is wrong. An boundless universe with uniform mass density has an infinite number of galaxies, particles, stars and planets in it and thus an infinite amount of matter.

Think of the universe like you would a 3D Cartesian coordinate system where a particle is located at the intersection of every grid point where a grid point is the point is of the forum (x, y, z) where x, y, z are all integers. Then the distance between all particles is finite yet the number of particles is infinite.

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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #20 on: 21/09/2014 01:36:48 »
Quote from: Bill
Pete, sometimes you surprise me.
Is that good or bad? In this post is it good or bad?  [:-\]

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Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #21 on: 21/09/2014 13:09:10 »
Quote from: jeffreyH
In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. So to all intents and purposes the contents of the universe is finite even though the universe itself may not be.
You're reasoning is wrong. An boundless universe with uniform mass density has an infinite number of galaxies, particles, stars and planets in it and thus an infinite amount of matter.

Think of the universe like you would a 3D Cartesian coordinate system where a particle is located at the intersection of every grid point where a grid point is the point is of the forum (x, y, z) where x, y, z are all integers. Then the distance between all particles is finite yet the number of particles is infinite.

Anything bounded cannot be infinite. The particles are bounded by an infinite extent.

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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #22 on: 21/09/2014 16:47:38 »
Quote from: jeffreyH
In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. So to all intents and purposes the contents of the universe is finite even though the universe itself may not be.
You're reasoning is wrong. An boundless universe with uniform mass density has an infinite number of galaxies, particles, stars and planets in it and thus an infinite amount of matter.

Think of the universe like you would a 3D Cartesian coordinate system where a particle is located at the intersection of every grid point where a grid point is the point is of the forum (x, y, z) where x, y, z are all integers. Then the distance between all particles is finite yet the number of particles is infinite.

Anything bounded cannot be infinite. The particles are bounded by an infinite extent.
Who said it was bounded? I don't know where on earth you're getting these ideas from but they sure aren't from me. You should know that I know that already. In fact  I'm the one who made that fact clear in the start of this thread!

I said think of the universe as you would a  3D Cartesian coordinate system. A  3D Cartesian coordinate system is unbounded, i.e. it's infinitely large, i.e. unbounded!

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #23 on: 21/09/2014 19:54:21 »
Quote from: Pete
Is that good or bad?

Assigning “good” or “bad” involves the sort of subjective value judgement I prefer to avoid in a science thread. 

Quote
In this post is it good or bad?   

That probably depends on whether you are asking this question to avoid answering the ones I asked.

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #24 on: 21/09/2014 20:11:07 »
Quote from: alancalverd
The gravitational force F exerted by an object of mass m decreases as m/r2, so F→0 as r→∞ . Physical reality? Well we can measure any  F > 0, so it's real 

Now the force exerted by a mass 2m decreases as 2m/r2, so F→0 as r→∞' and clearly ∞' > ∞
You are aware that my maths is shaky, so let’s be sure I understand what you are saying.

“The gravitational force F exerted by an object of mass m decreases as m/r2,”

Gravitational force decreases as a square of the distance over which it is measured.
 
“so F→0 as r→∞.”

Gravitational force would be measured as 0 only at an infinite distance from the source.

“Well we can measure any  F > 0, so it's real”

We can measure any force greater than 0, so the gravitational force is real.

Is my understanding OK so far?


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Offline Bill S

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Re: Is infinity a misconception?
« Reply #25 on: 21/09/2014 20:24:34 »
Quote from: Pete
You're reasoning is wrong. An boundless universe with uniform mass density has an infinite number of galaxies, particles, stars and planets in it and thus an infinite amount of matter.

Pete, I feel sure Jeffrey will correct me if I’m wrong here, but I don’t think this responds to the point he was making.

“….there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity.”

A particle must be somewhere.  Two particles must occupy two places.  If we say there is an infinite distance between these two places, surely, we are placing clear limits on infinity.  A distance that is limited in this way is clearly not infinite, so claiming that two objects can be at an infinite distance apart is a contradiction in terms. 

How is that reasoning wrong?

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #26 on: 21/09/2014 20:33:35 »
Quote from: Pete
A  3D Cartesian coordinate system is unbounded,

Yes.

Quote
i.e. it's infinitely large,

Only in principle.  We know that the Universe physically exists, within our understanding of physical existence, therefore to apply that assumption to the Universe is a leap too far.  It may be metaphysics, but it is not science.

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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #27 on: 21/09/2014 21:08:25 »
Is my understanding OK so far?

So far, so good. Now double the mass, so to measure any given value of F you have to stand √2 times as far away, so r tends to a different infinity as F tends to zero.   
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Offline Bill S

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Re: Is infinity a misconception?
« Reply #28 on: 22/09/2014 15:50:58 »
Quote from: alancalverd
  Now double the mass, so to measure any given value of F you have to stand √2 times as far away, so r tends to a different infinity as F tends to zero.

This is the sort of thing I’m looking for – a learning opportunity.

I understand how I would have to stand  √2 times as far away in order to measure any given value of F, but I do not see how this makes any difference to the infinity towards which r tends.   

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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #29 on: 22/09/2014 16:28:43 »
To experience the same value of F in both cases, you will have to stand further away in the second case. Consider a minute, teen weeny value of F. You will be 1.4 times further away from the source in the second case. So as F→0, r increases faster than in the first case. Therefore when F = 0, you are further away. But r1 → ∞, so r2 must tend to a greater infinity.

It isn't actually a problem, either in maths or physics. You can in principle have an infinite number of infinities, each depending on its definition, and all of various sizes.

Supose we have ten physical components, call them 0 to 9, which we can arrange in any sequence of any length - for instance integer numbers. There is an infinity of such possible combinations. Now suppose we have 26 components, called A to Z. Same rules - any sequence, of any length, e.g. Welsh compound words (no need for vowels!). Must be a bigger infinity!  So take an indefinitely extensible and branchable chemical chain like an aliphatic hydrocarbon: only two components and some strict rules about sequencing, but three dimensions and no limit on chain length: what size infinity describes the number of possible hydrocarbons?   
« Last Edit: 22/09/2014 16:42:00 by alancalverd »
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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #30 on: 22/09/2014 17:34:06 »
Quote from: Pete
You're reasoning is wrong. An boundless universe with uniform mass density has an infinite number of galaxies, particles, stars and planets in it and thus an infinite amount of matter.

Pete, I feel sure Jeffrey will correct me if I’m wrong here, but I don’t think this responds to the point he was making.

“….there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity.”
Let's take this one step at a time. I said ...that there's an infinite amount of hadrons in the universe ... in response to which Jeff replied In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. which is true. In fact that follows from uniform mass density. Jeff replied So to all intents and purposes the contents of the universe is finite even though the universe itself may not be. However that doesn't follow from what was said before that. That's why I gave him the example using 3D Cartesian coordinates as an example. There being a uniform mass density and no boundary on the universe, which means an infinite amount of space, means that there's an infinite amount of mass. Nothing personal folks but that's an extremely simple fact from algebra. Let u be the mass density, V the volume of the universe and M the mass of the universe. Then

M = u V

Since V is infinite and u finite it follows that M is also infinite.

Quote from: Bill S
A particle must be somewhere.  Two particles must occupy two places.  If we say there is an infinite distance between these two places,
which we aren't. That'd be impossible in itself.

Quote from: Bill S
How is that reasoning wrong?
See above.

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #31 on: 22/09/2014 19:15:34 »
Quote from: alancalverd
Therefore when F = 0, you are further away. But r1 → ∞, so r2 must tend to a greater infinity.

Consider what you are saying;

r1 → ∞, but it could never reach infinity.  However far it goes it is infinitely far from infinity. 

r2 → ∞, but it could never reach infinity.  However far it goes it is infinitely far from infinity.

In both cases you are infinitely far from infinity before you start, and when you finish.  Other than as a mathematical necessity, how does one infinity differ from the other?

Your reasoning is impeccable, as long as you consider infinity as a finite distance, which, manifestly it is not.

In this, and all of your examples, you are using mathematical infinities; I have no problem with that, and your arguments make perfect sense, as long as one remembers that mathematical infinities are approximations.

Interesting that you should mention Welsh words that have no need for vowels.  Let’s take a simple example, the word “pwll”; a Welshman looks at that and says: “lets call w a vowel”.  Now pwll has a vowel in it. 

Ar hyn sail, tybed os bob siaradwyr Cymraeg yn wyddonwyr.   [;D]

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #32 on: 22/09/2014 19:22:35 »
Thanks for that Pete.  It looks as though we had some crossed wires, but, to a great extent were on the same track.

If, as seems to be the case, you are saying it is impossible for two particles to be an infinite distance apart; I'm very happy with that.

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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #33 on: 22/09/2014 19:32:11 »
Quote from: Bill S
If, as seems to be the case, you are saying it is impossible for two particles to be an infinite distance apart; I'm very happy with that.
Absolutely since if two particles exist then they have a finite distance between then and infinite is not a distance.

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Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #34 on: 22/09/2014 20:43:53 »
Yes my flawed logic. Apologies Pete.

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #35 on: 22/09/2014 22:29:47 »
Quote from: JeffreyH
The point here is that any system that can normally be considered as bounded cannot include an infinite component.

Great!

Quote from: alancalverd
Precisely.

Terrific!!

Quote from: Pete
Let me make this very clear first; [infinity] is not a number.

It’s all coming together; but wait!  A little voice in the depths of my mathematical ignorance says:  “What about the interval from 1 to 2?  This is bounded on both sides by an integer, yet - 

Quote from: alancalverd
  Indeed there is an infinite number of rational numbers between any two integers.

Does it all depend on what we decide that “infinite” should mean?"

Take consolation from the fact that AC got the number of his verb to agree with that of the noun with which it is construed.  That’s more than most people seem to be able to manage in that sort of sentence.    [;)]
« Last Edit: 22/09/2014 22:49:00 by Bill S »

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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #36 on: 22/09/2014 22:54:40 »


r1 → ∞, but it could never reach infinity.
obviously, by definition. 

Quote
r2 → ∞, but it could never reach infinity.
also true, but as it is increasing more quickly than r1 it must at all times be greater than r1 and increasingly so, which means the inifinity iit is tending to must be larger than that of r1.

Quote
Your reasoning is impeccable, as long as you consider infinity as a finite distance, which, manifestly it is not.
No, it's only impeccable if you understand that there are different infinities.

Quote
In this, and all of your examples, you are using mathematical infinities; I have no problem with that, and your arguments make perfect sense, as long as one remembers that mathematical infinities are approximations.
Not at all. The definition of any infinity is absolutely precise. Take the simplest infinity: 1/x where x→0. x=0 is an absolutely precise statement, not an approximation to anything.

As for Welsh, there must surely be an infinity of words if any letter can be considered a vowel and any consonant can be pronounced in any way as long as it doesn't sound like English.  The clever bit is that they all mean "the beautiful sadness of the oppressed". Or was my Welsh neice lying about the song she sang at the last Eisteddfodd?
« Last Edit: 22/09/2014 22:56:52 by alancalverd »
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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #37 on: 22/09/2014 23:10:43 »
Quote from: JeffreyH
The point here is that any system that can normally be considered as bounded cannot include an infinite component.

It’s all coming together; but wait!  A little voice in the depths of my mathematical ignorance says:  “What about the interval from 1 to 2?  This is bounded on both sides by an integer, yet - 

Quote from: alancalverd
  Indeed there is an infinite number of rational numbers between any two integers.



Not a problem. A bounded set can contain anything and everything that can fit between the bounds. But the number of objects in the set is not necessarily a member of the set. Consider how many cats can sit on a roof. The number of cats is not a cat, and is not located on the roof.

Now the number of numbers in a set is an integer but not necessarily a member of the set. Consider the number of integers between 11 and 16: it is 6, which is not between 11 and 16. and the number of quarters between those bounds is 64, which is also not a member of the set, so the number of objects in a denumerable set need not be a member of that set. Rational numbers are denumerable (you can write them all down in sequence and count them, and you can write a recursive algorithm for generating them) but there is an infinite number of rational numbers in any interval.       
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Offline evan_au

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Re: Is infinity a misconception?
« Reply #38 on: 23/09/2014 12:54:51 »
A radical idea:
Rather than teaching young kids arithmetic on the numbers 1 to 12, how about starting by teaching them arithmetic on the 3 numbers 0, 1 and ∞?

The arithmetic on these 3 is somewhat simple; a syllabus might look something like:
Addition (can be viewed as repeated counting up)
  • n+0=n
  • n+1=next larger countable number (leads to concept of >; statistics can be viewed as counting outcomes)
  • n+∞=∞
  • n+m=m+n
Subtraction (Can be viewed as repeated counting down)
  • n-0=n
  • n-1=next smaller countable number (leads to concept of <)
  • 0-n=-n (leads to negative numbers)
  • n-∞=-∞
  • ∞-n=∞
Multiplication (can be reviewed as repeated addition)
  • n*0=0
  • n*1=n
  • n*∞=∞
  • n*m=m*n
Division (can be viewed as repeated subtraction)
  • 0/n=0
  • n/1=n
  • n/∞=0
  • n/0=∞ (introduces ∞, after dealing with 1 & 0)
  • ∞/n=∞
  • 1/n= (introduce fractions)
  • 0/0= need more information (leads to limits as n→0, plus differentiation & integration; add L'Hopital's rule in senior maths)
  • ∞/∞= need more information (leads to limits as n→∞, plus differentiation & integration; add L'Hopital's rule in senior maths) 
For precise answers, use a calculator; humans should use estimation.
Later introduce square roots (via Pythagoras leads to geometry and trigonometry; irrational numbers  & imaginary numbers).
Also introduce exponention (which can be viewed as repeated multiplication; leads to Cantor's hierarchy of infinities in university).
« Last Edit: 24/09/2014 23:25:08 by evan_au »

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #39 on: 23/09/2014 14:12:02 »
Quote from: alancalverd
The clever bit is that they all mean "the beautiful sadness of the oppressed". Or was my Welsh neice lying about the song she sang at the last Eisteddfodd?

Your niece was not lying, but there is so much more to Welsh speaking than sadness and oppression.  It embodies the strength and determination of a people moving forward and taking their rich tradition with them.


I have no problem with the number of cats you may have on your roof, nor would I dispute the intricacies of set theory.  What I have to ask is: Are you saying that Jeffrey was wrong when he said that “any system that can normally be considered as bounded cannot include an infinite component.”? 

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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #40 on: 23/09/2014 15:29:06 »
Quote from: jeffreyH
Yes my flawed logic. Apologies Pete.
No problem my dear Jeff. What I admire about you is your astute ability to both recognize your mistakes and admit them. Something a lot of people don't have the ability to do.

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #41 on: 23/09/2014 17:11:45 »
Quote from: evan
n+∞=∞

That’s OK as long as you are talking about mathematical infinities.  If you are talking about an infinite cosmos, then n+∞=∞ has no real meaning, because the infinite cosmos is all that exists, or can exist; in which case there is no 1 to add to infinity.

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #42 on: 23/09/2014 17:18:26 »
Quote from: alancalverd
Not at all. The definition of any infinity is absolutely precise. Take the simplest infinity: 1/x where x→0. x=0 is an absolutely precise statement, not an approximation to anything.

This is probably a very naïve question, but is x→0 the same as x→ ∞?

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Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #43 on: 23/09/2014 18:41:00 »
Quote from: jeffreyH
Yes my flawed logic. Apologies Pete.
No problem my dear Jeff. What I admire about you is your astute ability to both recognize your mistakes and admit them. Something a lot of people don't have the ability to do.

Reasoned debate is more productive than unreasoned bile. I would rather learn from mistakes than keep repeating them. I often spout nonsense but it takes me a while to realize it.

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Offline Bill S

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Re: Is infinity a misconception?
« Reply #44 on: 23/09/2014 21:41:45 »
Quote from: evan
n-∞=-∞

I've been trying to get my head round that one.  How do you define minus infinity?  It's a fascinating thought, but it beats me.   [???]

Alright, already! I know that's easy when it comes to maths.

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Offline JohnDuffield

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Re: Is infinity a misconception?
« Reply #45 on: 23/09/2014 22:16:54 »
That’s OK as long as you are talking about mathematical infinities.  If you are talking about an infinite cosmos, then n+∞=∞ has no real meaning, because the infinite cosmos is all that exists, or can exist; in which case there is no 1 to add to infinity.
We just don't know that the universe is infinite, Bill. I don't think it can be, because the universe has a finite age, I do not accept that it was already infinite when the big bang occurred, and I don't see how an infinite universe can possibly expand because the "pressure" is counterbalanced at all locations.

Quote from: Bill S
This is probably a very naïve question, but is x→0 the same as x→ ∞?
No. We have plenty of instances where something diminishes to zero, but we have no evidence that there are any infinities in nature. When they crop up, such as with a black-hole point-singularity, they are thought to signify some breakdown in the mathematics and in our understanding.
« Last Edit: 24/09/2014 08:37:23 by JohnDuffield »

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Offline alancalverd

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Re: Is infinity a misconception?
« Reply #46 on: 23/09/2014 23:06:45 »
I have no problem with the number of cats you may have on your roof, nor would I dispute the intricacies of set theory.  What I have to ask is: Are you saying that Jeffrey was wrong when he said that “any system that can normally be considered as bounded cannot include an infinite component.”? 


Yes, he was wrong. Hence my example that there is an infinity of rational numbers in any interval. The interval between 0 and 1 contains 1/2, 1/3, 1/4....2/3, 2/4, 2/5,....3/4, 3/5, 3/6....and so on - i.e. an infinite number of rationals exist between the bounds of 0/1 and 1/1. Whilst the number of rationals in an interval is not a component of that interval, there is an infinite number of components, so the set of components is infinite and thus the bounded interval contains an infinite component.
helping to stem the tide of ignorance

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Offline evan_au

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Re: Is infinity a misconception?
« Reply #47 on: 23/09/2014 23:07:39 »
Quote from: Bill S
is x→(Finite number) the same as x→∞?
It's not the same, but the two are often related.
For example, relativity has many expressions like 1/√(1-v2/c2)

As v→c, 1/√(1-v2/c2) →∞.

This relationship has been validated to many decimal places in successively higher energies of particle accelerators.

JohnDuffield points to another example in a black-hole point-singularity, but in that case it is likely that the quantum nature of space and particle tunneling will provide a lower limit on the degree to which the inverse square law applies as distance x→0 (but we currently have no tested theory of quantum gravity).

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Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #48 on: 23/09/2014 23:37:50 »
I have no problem with the number of cats you may have on your roof, nor would I dispute the intricacies of set theory.  What I have to ask is: Are you saying that Jeffrey was wrong when he said that “any system that can normally be considered as bounded cannot include an infinite component.”? 


Yes, he was wrong. Hence my example that there is an infinity of rational numbers in any interval. The interval between 0 and 1 contains 1/2, 1/3, 1/4....2/3, 2/4, 2/5,....3/4, 3/5, 3/6....and so on - i.e. an infinite number of rationals exist between the bounds of 0/1 and 1/1. Whilst the number of rationals in an interval is not a component of that interval, there is an infinite number of components, so the set of components is infinite and thus the bounded interval contains an infinite component.

Physically you cannot go on sub-dividing space. You hit the Planck scale before you know it. As far as maths goes you are right but you soon run into a Zeno's paradox at very small physical scales.

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Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #49 on: 24/09/2014 14:29:10 »
JohnDuffield is a major crackpot - Beware!!!!