[PmbPhy]

Pete, sometimes you surprise me.

Is that good or bad? In this post is it good or bad?  []

[jeffreyH (OP)]

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.

[PmbPhy]

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!

[Bill S]

Is that good or bad?

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

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.

[Bill S]

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?

[Bill S]

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?

[Bill S]

A  3D Cartesian coordinate system is unbounded,

Yes.

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.

[alancalverd]

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.   

[Bill S]

  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.   

[alancalverd]

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?   

[PmbPhy]

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.

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.

How is that reasoning wrong?

See above.

[Bill S]

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.   []

[Bill S]

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.

[PmbPhy]

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.

[jeffreyH (OP)]

Yes my flawed logic. Apologies Pete.

[Bill S]

The point here is that any system that can normally be considered as bounded cannot include an infinite component.

Great!

Precisely.

Terrific!!

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 - 

  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.    []

[alancalverd]

r1 → ∞, but it could never reach infinity.

obviously, by definition. 

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.

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.

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?

[alancalverd]

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 - 

  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.       

[evan_au]

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).

[Bill S]

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.”?