Are you aware of a reason, in your own mind, why you believe you are posting these things?

Any thoughts on the horn?

My thought is, don't sit on it.

This one is a bit trickier and as far as I know it has not been resolved. So if we can't get it off the ground, I have others waiting.

Thank you for your response.

My thought is, don't sit on it. — Baden

Wise advice.

Fortunately, it is impossible to sit on it, because it has no tip. The pointy bit just recedes endlessly, never culminating in a spike. The ultimate in child-safety mathematical structures.

As for getting it off the ground, that would be impossible because, even though it has finite volume, and hence finite mass (if we assume constant density), its moment of inertia would be infinite because of its being infinitely long. So it would require an infinite torque to rotate it to an erect position.

Short version - funny things happen with infinity. (one reason why maths is so much fun)

Unlike the horn, my post had a real point. :) That was informative and fun to read though. :up:

So how it is possible this horn can have limited volume but endless surface area? — Jeremiah

In mathematics, any volume can be divided in such a way to cover any surface.

This is just like Zeno's paradox. The paradox arises from the confusion of abstract properties with real ones of the same name.

So you are suggesting if it was filled with paint, you could use a finite amount of paint to paint an endless surface.


It seems to me, that you'd run out of paint, and even if you could stretch the paint infinitely thinner, that still does not resolve the paradox. As abstractly what you have is a cone with a converging volume and a diverging surface area.

So you are suggesting if it was filled with paint, you could use a finite amount of paint to paint an endless surface. — Jeremiah

It is trivial to divide any volume to cover an infinite surface. There are plenty of convergent infinite series that will divide the volume for you.

It seems to me, that you'd run out of paint, and even if so that still does not resolve the paradox. As abstractly what you have is a cone with a converging volume and a diverging surface area. — Jeremiah

Like I said. You are confusing abstract and physical properties that happen to have the same name.

This one is a bit trickier and as far as I know it has not been resolved. — Jeremiah

In what way is this in need of 'resolution'? You haven't stated a problem with this scenario.
Is there some law somewhere being broken, like infinite surfaces must enclose infinite space? There is obviously no such law, as demonstrated by this example.

So you are suggesting if it was filled with paint, you could use a finite amount of paint to paint an endless surface.

It seems to me, that you'd run out of paint, and even if you could stretch the paint infinitely thinner, that still does not resolve the paradox. As abstractly what you have is a cone with a converging volume and a diverging surface area. — Jeremiah

Clearly the paint would not run out, as it hasn't in your example. It covers the entire surface, and doesn't even need to be spread out to do so, since it has finite thickness (all the way to the center line) at any point being painted.

I see no paradox in need of resolution. The volume converges and something different (the area) does not. It is only paradoxical if the same thing both converges and diverges.

In what way is this in need of 'resolution'? You haven't stated a problem with this scenario. Is there some law somewhere being broken, like infinite surfaces must enclose infinite space? There is obviously no such law, as demonstrated by this example. — noAxioms

The paradox, seems clear to me, we have a container that stretches on forever, yet it has a finite volume.

It is only paradoxical if the same thing both converges and diverges. — noAxioms

The horn both converges and diverges, so it fits your personal take on what is needed for a paradox.

Clearly the paint would not run out, as it hasn't in your example. It covers the entire surface, and doesn't even need to be spread out to do so, since it has finite thickness (all the way to the center line) at any point being painted. — noAxioms

So you are suggesting a finite amount of paint that goes on forever. So in your suggestion the volume of the paint both converges and diverges? Well, mathematically we can prove the volume of the paint converges, that means there is a limited amount of it, but if you want to claim it is a endless bucket of paint go for it. The math just does not back you up.

Like I said. You are confusing abstract and physical properties that happen to have the same name. — tom

If you recall I never said or agreed to any such notion in the last thread. I avoid that line of thought for a reason. There is nothing which says we can't think about this in more practical terms.

Any container or solid object that has an endless surface area, but a finite volume is paradoxical, abstractly or otherwise.

Volume is the amount of space it takes up, so if it has endless surface area it should have endless volume. However, Gabriel's horn does't, and this is why it is widely recognized as a paradox.

Any container or solid object that has an endless surface area, but a finite volume is paradoxical, abstractly or otherwise. — Jeremiah

No such object can exist in Reality, so it cannot be "abstractly or otherwise".

Gabriel's horn exist in reality, the math was posted in the OP.

Abstract things exist in reality, as reality is a very very very broad term.

Claiming it is abstract does't prove that Gabriel's horn is not a mathematical conundrum.

The horn both converges and diverges, so it fits your personal take on what is needed for a paradox. — Jeremiah

A paradox is usually of the form of "If A is true, then A can be shown to be false". Your original 25 25 50 60 thingy would have been paradoxical had the 60 entry read 0%. What you seem to be reaching for here is not a paradox, but rather a violation of the law of non-contradiction, that a thing cannot be both X and not-X at the same time in the same way. I don't see the violation due to the 'in the same way' part.

So you are suggesting a finite amount of paint that goes on forever.

that paints an infinite surface. 'goes on forever' is not what I said, and seems a sort of undefined wording.
The alternative is that there is some points along your surface that do not enclose volume and are thus not painted.

So in your suggestion the volume of the paint both converges and diverges?

No, the volume is finite. You said that. There is finite (convergent as you put it) volume of ice cream, which could be paint.

Any container or solid object that has an endless surface area, but a finite volume is paradoxical, abstractly or otherwise.

Volume is the amount of space it takes up, so if it has endless surface area it should have endless volume. — Jeremiah

This assertion is exactly that: just an assertion, and a false one at that. There is no mathematical basis for this. The paradox apparently comes from your assumption of this nonexistent law.

Well this is not my paradox, I didn't invent it. It is a well known paradox, and widely recognized as such. Also the mathematical proof is posted in the OP. Saying there is no mathematical basis for this just tells me you can't read the math, as it is posted right there for you to review.

I don't really see you as an authority on what is and what is not a paradox. I mean all you have here is an assertion and a false one at that. On the other-hand academically Gabriel's horn is widely viewed as paradoxical. So you don't think is a paradox, OK fine, I don't really care.

Also volume is the amount of space an object takes up, paint or no paint.

Maybe this one is harder for people to sink their teeth into.

Gabriel's horn is an object that exist in math which has finite volume, but infinite surface area, that is a conundrum if I have ever seen one. Such objects should not exist, but mathematically we can show that the volume converges to a finite point, while the surface area diverges to infinity.

You can say, well it is not in the real world, and while it may be true I can't find a horn and point to it; however, it does exist in mathematics, and this is the math section of these forums and the title of this thread is "Mathematical Conundrum or Not?". The horn absolutely deserves its spot here, even if grasping it is not as intuitive as the other two I posted.

Well this is not my paradox, I didn't invent it. It is a well known paradox, and widely recognized as such. Also the mathematical proof is posted in the OP. — Jeremiah

So you don't think is a paradox, OK fine, I don't really care. — Jeremiah

Fair enough. The relevant definition of paradox that pops up says this:

a : a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true
b : a self-contradictory statement that at first seems true
c : an argument that apparently derives self-contradictory conclusions by valid deduction from acceptable premises — Webster

The funny cone seems to fall under definition 'a' since it seems opposed to common sense to many people. So yes, it makes sense to 'resolve' such paradoxes by showing that the seeming contradiction is something that is actually the case. The mathematics (a computation of the area and volume) is linked in the OP, but not sure what part of that is a 'proof' of something.

'b' seems to be the opposite of 'a': something that seems true at first but false on closer inspection.

I guess my idea of a paradox falls under 'c', the most basic example being "This statement is false". Any truth value assigned to that seems to be incorrect. I've seen it resolved in law-of-form using an imaginary truth value (square root of false) just like imaginary numbers solve square root of -1. There is application for such logic in quantum computing.

Saying there is no mathematical basis for this just tells me you can't read the math, as it is posted right there for you to review.

I never contested the mathematics, which simply shows that the object indeed has infinite area but finite volume. I can think of more trivial objects that are finite in one way but infinite in another, and your cone did not strike me as a connundrum. But I retract my assertion that it is not a paradox. The definition above speaks.

You can say, well it is not in the real world — Jeremiah

Indeed, it is only a mathematical object. A real one could not be implemented, growing too thin to insert ice cream particles after a while.

Interestingly, a liter of physical paint contains insufficient paint to actually cover a square meter of surface. There is a finite quantity of fundamental particles making up the volume of paint, and no fundamental particle has ever been found that occupies actual volume. So the paint is all empty space with effectively dimensionless objects which are incapable of being arranged to cover a given area without gaps. Instead, paint atoms work by deflecting light and water and such using its EM properties, not by actually covering a surface without gaps. Point is that this particular mathematical object has little relevance to even a hypothetical physical object.

It's not quite the same thing, but this reminds me of the story of how Dido bought the land that became Carthage by agreeing to buy as much land as she could encompass with a single oxhide. By cutting the hide into extremely thin strips, she was able to section off quite a bit more than the sellers reckoned with.

She wasn't working with infinite amounts, but I wonder if there's a way to figure you could encompass an infinite area with a finite mass?

Such objects should not exist — Jeremiah

That is a feeling. The 18th century British invaders of Australia had a similar feeling when they first saw a platypus. When they found that the object in question was undeniably there in front of them, their 'should not exist' transformed to 'well, I am very surprised'.

Is the aim of this thread then to muse over the nature of the emotion we call Surprise?

Let me know if you figure it out.

It's not for me to figure out. You started the thread. What was your aim?

I'll give you a hint, it has nothing to do with a platypus.

That's the third time you dodged the question - which was originally put to you in post #2. Are you going to answer the question? What was your aim?