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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?

The notationmeans 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.

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

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.

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

Let me make this very clear first;is not a number.

Let me make this very clear first; is not a number.

IMHO if ever you bump into an infinity in physics, then something is wrong somewhere.

Quote from: JohnDuffieldIMHO 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.

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.

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

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.

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

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.

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.

Pete, sometimes you surprise me.

Quote from: jeffreyHIn 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.

Quote from: PmbPhy on 21/09/2014 01:35:20Quote from: jeffreyHIn 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.

Is that good or bad?

In this post is it good or bad?

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'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.

A 3D Cartesian coordinate system is unbounded,

i.e. it's infinitely large,

Is my understanding OK so far?

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.

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,

How is that reasoning wrong?

Therefore when F = 0, you are further away. But r1 → ∞, so r2 must tend to a greater infinity.

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.

Precisely.

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

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

r1 → ∞, but it could never reach infinity.

r2 → ∞, but it could never reach infinity.

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.

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.

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?

Yes my flawed logic. Apologies Pete.

n+∞=∞

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.

Quote from: jeffreyHYes 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.

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.

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

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.”?

is x→(Finite number) the same as x→∞?

Quote from: Bill S on 23/09/2014 14:12:02I 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.