[jeffreyH (OP)]

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?

[alancalverd]

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.

[jeffreyH (OP)]

Well then find the radius of the circle whose circumference is infinite.

[jeffreyH (OP)]

Conversely if the radius is infinite find the circumference.

[jeffreyH (OP)]

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

[PmbPhy]

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; is not a number.

It's defined as follows;

The notation



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 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

[alancalverd]

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.   

[JohnDuffield]

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?



IMHO it's better to say that the coordinate speed of light is zero at the event horizon.

[Bill S]

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.

[alancalverd]

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.....

[PmbPhy]

“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

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

Your quote instead reads

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

[PmbPhy]

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.

[jeffreyH (OP)]

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.

[JohnDuffield]

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.

[Bill S]

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. 

[Bill S]

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.

[Bill S]

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”.   

[Bill S]

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.

[alancalverd]

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 ∞' > ∞

[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.