The physical properties of the paint being incompatible with an infinite horn, was already rejected as not the subject of this discussion. If we were discussing whether the molecules of paint could fit down inside an infinitely small tube, we might just as well reject the infinitely small tube as a nonsensical proposition in the first place. — Metaphysician Undercover

See? You don't even know what you're discussing!

Andrewk does not provide a solution. The inside of the horn has a non-zero diameter with infinite extension. This means that there is an infinite surface area on the inside of that horn, to be covered with paint, just like there is on the outside. — Metaphysician Undercover

I don't see how this is relevant. Are you forgetting that it's infinitely long? — Metaphysician Undercover

so whatever the layer is, it will be multiplied by infinity, — Metaphysician Undercover

That is what we're discussing. You made a claim that there's some number, you call it "whatever the layer is", that you multiply by infinity. The problem is, there's only a finite portion of the horn that can fit "whatever the layer is", and there's an infinite portion of the horn too thin to have "whatever the layer is" be the thickness of "whatever the layer is". Because of this, you cannot multiply "whatever the layer is" by infinity and get anything meaningful.

You're the one who keeps dragging physical properties of paint into this. I'm just talking about your wrongness; your "whatever the layer is" that you think you get to multiply by infinity.

We could be having the same argument about your ability to fit a y by y by x rectangular prism in the center of infinite volume (V=y*y*x, with x being "infinitely long"). There's no such prism you can fit in the horn; no matter what your y is, it will only go in so far. The only way you get to have an infinitely long prism in there is if y is 0; and 0*0*infinity is indeterminate.

The problem is, there's only a finite portion of the horn that can fit "whatever the layer is", and there's an infinite portion of the horn too thin to have "whatever the layer is" be the thickness of "whatever the layer is". Because of this, you cannot multiply "whatever the layer is" by infinity and get anything meaningful. — InPitzotl

Obviously, if we can assume that the horn is infinitely long, with an infinitely small diameter, we can also assume that the paint can go infinitely thin. That this is not an issue of the real physical properties of a real horn, or the real physical properties of real paint was determined in the first couple of replies in the thread. Do you agree that if the horn is allowed to go infinitely thin, then the paint must play by the same rules, and be allowed to go infinitely thin as well?

Obviously, if we can assume that the horn is infinitely long, with an infinitely small diameter, we can also assume that the paint can go infinitely thin. — Metaphysician Undercover

You're kind of mixing two things in here. Imagine a mathematical bag; inside the bag, we'll put all finite numbers. All of them, mind you, but only the finite ones. There are an infinite number of numbers in this bag, but infinity isn't in the bag. So if we admit that the horn is infinitely long, we're not necessarily admitting that there's an infinity-point on the horn (a point where there's an infinitely small diameter).

But if you want to talk about an infinite point on the horn, you're free to do so (it's just not implied by admitting the horn is infinitely long). You can even say the horn is infinitely thin at that point. But that doesn't help you here:

so whatever the layer is, it will be multiplied by infinity, — Metaphysician Undercover

...because now you're talking about an infinitely tiny quantity multiplied by infinity. And that's still not meaningful.

Do you agree that if the horn is allowed to go infinitely thin, then the paint must play by the same rules, and be allowed to go infinitely thin as well? — Metaphysician Undercover

I already discussed that here.

I already discussed that here. — InPitzotl

At that point you're basically just mapping points to points, and there are plenty enough points in a tiny droplet of paint to map to the infinite surface area of the horn. — InPitzotl

In other words you want the paint to follow different rules for the inside of the horn, than for the outside of the horn, allowing a finite volume of paint to cover an infinite surface area on the inside, but not the outside. What's the point, if you can just make up whatever rules you want?

Why can't we just do the same thing with the infinite surface area on the outside of the horn as well? We can just say that any tiny amount of paint can be spread out over the infinite outside surface area, just like you say for the inside.

In other words you want the paint to follow different rules for the inside of the horn, than for the outside of the horn, allowing a finite volume of paint to cover an infinite surface area on the inside, but not the outside. — Metaphysician Undercover

Wrong. You're only confusing yourself here. I haven't specified any rules for paint at all, much less different rules for the inside and outside. Rather, I've talked about three things:

(A) Intuitive paint, which is based on real life paint (not necessarily physics; a painter can get by just fine if he believes atomic theory is a conspiracy, so long as he follows the rules of intuitive paint whereby he calculates he needs 8 gallon cans to paint 3000 square feet).
(B) tim wood's paint, which he explicitly said wasn't intuitive paint and must meet particular conditions
(C) MU's paint, a strange kind of thing that supposedly lets us multiply some number by infinity and get some meaningful conclusion out of it

(A) is where all of the confusion sets in; it's why we think there's a paradox when there is in fact nothing here. When I use my gallon of paint to paint a 20 foot by 20 foot square, and I think that the thing I painted was an area, I've just tricked myself into thinking volumes (gallons) relate to areas (square feet); namely, that 1 gallon=1/7.5 cubic feet relates to 400 square feet. But that intuitive relation is illusory; the 20x20 swatch is actually 1/3000 foot thick, making that paint layer a volume not an area. Gabriel's horn has infinite surface area, but holds a finite volume. But as I've said repetitively, areas have no volume. "An infinite number of square feet" conveys no meaningful amount of cubic feet.

(B) is some new kind of paint tim wood was proposing; whatever that was, it's something that by filling the horn counts as painting its inside. That's the discussion I linked you to. But that sort of paint necessarily must count a width 0 layer of paint (or if you prefer, infinitely thin) as painting the inside surface. There's no separate rule for the outside; (B) kind of paint is just as good for the outside as the inside, and you don't even need pi paint to paint either... any tiny droplet would do. In fact, any volume of paint would paint any area, even if the volume is small and the area infinite.

Then we have C paint, whereby MU is trying to justify that multiplying some number by infinity means that volumes really do relate to areas, and/or demonstrate that he should not have to understand a conversation before he pretends he's contributing to it.

Gabriel's horn has infinite surface area, but holds a finite volume. — InPitzotl

The volume of the horn is only determined as finite when the infinite radius is rounded off to zero at some determinable length, as is demonstrated by the YouTube video. Otherwise the infinite length of the horn ensures that the volume is infinite.

The volume of the horn is only determined as finite when the infinite radius is rounded off to zero at some determinable length, as is demonstrated by the YouTube video. — Metaphysician Undercover

Wrong. The video used limits (and integrals, which are built off of limits). Limits don't round off to zero.

Here's how a limit works:
$\lim_\limits{x \rightarrow \infty}{\frac{1}{x}}=0$
...and it's equal to 0 exactly... not rounded off, because L=0 meets the conditions:
$\lim_\limits{x \rightarrow \infty}{\frac{1}{x}}=L \Rightarrow \forall \epsilon \gt 0, \exists M : x \gt M \Rightarrow L - \epsilon \lt \frac{1}{x} \lt L + \epsilon$
...and that can be shown generically for any ϵ. For such ϵ, simply choose $M=\frac{1}{\epsilon}$.

No other number works for that limit; a billionth doesn't work for example; ϵ=two billionths betrays it, because for all x's greater than two billion, $\frac{1}{x} \lt \frac{1}{2,000,000,000}$ which is more than two billionths away from one billionth.

Likewise, every close to 0 integer doesn't work; only 0 exactly works.

Incidentally, integration and limits were used in that video, so this:

Clearly you are using a different calculation than the one in the video then. If you know of a method to figure out the volume of that horn, which avoids rounding off the infinitely small dimeter to zero, then maybe you should present it for us. — Metaphysician Undercover

...is just uninformed non-sense.

Sure chief, one over infinity is zero, and that's not a matter of rounding off. Tell me another.

If you know of a method to figure out the volume of that horn, — Metaphysician Undercover

Here's how a limit works:
$\lim_\limits{x \rightarrow \infty}{\frac{1}{x}}=0$ — InPitzotl

Sure chief, one over infinity is zero, and that's not a matter of rounding off. — Metaphysician Undercover

At this point, it's just denial, and you're unqualified to continue this discussion with. Hardly surprising, given this is the same exact thing you failed to grasp in the other thread.

LOL! Look who's in denial!
Saying that 1/infinity equals zero is obviously an instance of rounding off. There's nothing wrong with rounding off. We do it all the time with pi, square roots, etc.. That's how we get the job done by rounding off. If we couldn't round off, we couldn't get the job done in many instances. So you shouldn't be embarrassed by it. You should be embarrassed by insisting that it's not an instance of rounding off, when it clearly is, though.

Saying that 1/infinity equals zero is obviously an instance of rounding off. — Metaphysician Undercover

It's right there under your nose and you can't see it. You read:
$\lim_\limits{x \rightarrow \infty}{\frac{1}{x}} = 0$
...as "saying 1/infinity equals 0". But that's not what it says, and it's not what it means. I told you what it means, and showed you a link.

There's nothing wrong with rounding off. We do it all the time with pi, square roots, etc. That's how we get the job done by rounding off. If we couldn't round off, we couldn't get the job done in many instances. So you shouldn't be embarrassed by it. — Metaphysician Undercover

Okay, so there's nothing wrong with rounding off. And okay, we do it all of the time. So what? It's cute and all that you're trying to "counsel" me so that I can "cope" with rounding off, but your projection of some imagined psychological trauma is a red herring. Limits still aren't rounding off.

You should be embarrassed by insisting that it's not an instance of rounding off, when it clearly is, though. — Metaphysician Undercover

Clearly it's not rounding off, though. Clearly, you just don't understand what a limit is. And that's okay, MU. Not understanding something isn't the end of the world. It's nothing to be embarrassed about; there's a lot of knowledge in the world and not everyone knows everything. There's nothing to be embarrassed about by admitting that you don't know something. But you should be embarrassed by insisting that you understand when, clearly, you don't.

Rounding off implies that there's a stated answer a, and a real answer b, and that a is not b but is "close enough" to it.

But there's no such thing as an $L \neq 0$ such that:
$\forall \epsilon \gt 0, \exists M : x \gt M \Rightarrow L - \epsilon \lt \frac{1}{x} \lt L + \epsilon$
There is, however, such a thing as an L such that this condition is met; namely, L=0.

So the limit here is met by the value 0 exactly. This is a binary thing; either something works, aka meets the definitive criteria, or it doesn't work, aka it doesn't meet the criteria. 0 falls in the "meets the criteria" camp; it's the real answer b. "0 approximately" falls in the doesn't meet the criteria camp. There's genuinely no rounding off here. There is only an infinite amount of MU confusion.

He says "well one over infinity that's zero, so you get nothing from that". — Metaphysician Undercover

t's right there under your nose and you can't see it. You read:
limx→∞1x=0limx→∞1x=0
...as "saying 1/infinity equals 0". But that's not what it says, and it's not what it means. I told you what it means, and showed you a link. — InPitzotl

I saw it, and heard it distinctly stated, on the Youtube video at the referenced point, he distinctly says "well one over infinity that's zero, so you get nothing from that". And, you said: your method is the same as the one on the video. You can insist, as fishfry stated, that this is the convention in such procedures, to take one divided by infinity as zero. But that is to round off, so we need to respect the fact that the solution to the question of the volume of the horn, is a rounded off solution.

Limits still aren't rounding off. — InPitzotl

"Infinite" means unlimited. When you apply limits to the unlimited, you are either contradicting or rounding off. Which charge do you prefer? If you will not accept the fact that you are rounding off, then the paradox arises due to the contradiction.

Rounding off implies that there's a stated answer a, and a real answer b, and that a is not b but is "close enough" to it. — InPitzotl

Yes, and that describes exactly what is the case with the volume of Gabriel's horn. The "real answer" is that the horn is, stipulated by the stated premise, as infinitely long. And, the volume of an infinitely long container cannot be determined. The "close enough" answer is produced by applying the limits of integration, which works with approximations. The appearance of a paradox arises, because "close enough" fails the task intended by the proposition "infinite". In other words, there is no such thing as "close enough" when we're talking about the infinite.

So the limit here is met by the value 0 exactly. — InPitzotl

To impose a limit on the infinite is to contradict. So it is not the determined value, 0, which is wrong. That value is correct according to the terms of application, but the judgement that the situation is suited to the application, is wrong. The example stipulates that the horn is of infinite length. Therefore to impose a limit on its length is a mistaken procedure.

In common practice, this is not a problem. We don't often encounter infinitely long things which we want to figure the volume of. But when we encounter an infinitely long thing proposed in a thought experiment, it is of paramount importance that we stay true to the premises, and have one consistent way of interpreting "infinite", or else someone will claim that the results are paradoxical.

It's seems quite simple to take the easy way out and suggest the divine link to the infinite - archangel Gabriel trumpet to announce Judgement Day and the finite number that the divine will take or make it through Judgement...there's not a paradox as such that brings doubt and uncertainty, there is instead the finite within the infinite and vice versa the infinite in the finite...the concept of the divine is difficult to grasp or imagine because it goes beyond our imagination as finite humans...those who believe in a soul understand that this is a connection with the infinite within the finite...as for paint that's another substance that has a finite life cycle....
The mathematics others have presented ...still complex and fascinating and obviously I haven't watched the YouTube clip...

I saw it, ... he distinctly says "well one over infinity that's zero, so you get nothing from that". — Metaphysician Undercover

But you didn't understand it.

And, you said: your method is the same as the one on the video. — Metaphysician Undercover

Yes, but the method is integration.

You can insist, as fishfry stated, that this is the convention in such procedures, to take one divided by infinity as zero. — Metaphysician Undercover

But that is not the method; that is just a shortcut. The method is to apply a limit.

"Infinite" means unlimited. — Metaphysician Undercover

Sure. But remember the bag I was talking about earlier? That bag has all of the finite numbers in it, but no infinite numbers. This is the situation here. The bag has real numbers in it; there's no such thing as an infinite real number though. Nevertheless, "unlimited" describes the extent of such numbers on the number line in the positive direction. And that ∞ symbol when used in the limit is used to represent just that... that's not a number, it's just shorthand for representing the infinite extent of the numbers in my bag.

For example, a lower limit of 1 and an upper limit of 2 refers to "all of the numbers in the bag that are greater than or equal to 1, and less than or equal to 2". By contrast, a lower limit of 1 and an upper limit of ∞ simply means: "all of the numbers in the bag that are greater than or equal to 1".

When you apply limits to the unlimited, you are either contradicting or rounding off. — Metaphysician Undercover

Nonsense. You just made that up.

If you will not accept the fact that you are rounding off, then the paradox arises due to the contradiction. — Metaphysician Undercover

Those things don't follow, there isn't a paradox here in the first place, and anyone trying to play the if-you're-wrong-that-means-I'm-right card should have their philosophy license revoked.

that describes exactly what is the case with the volume of Gabriel's horn — Metaphysician Undercover

If you don't understand the method, you're unqualified to critique it.

The "real answer" is — Metaphysician Undercover

Wrong. The real issue is very simple... areas aren't volumes. "Paint" tricks you into thinking they are. It is very interesting to note that you never actually tried to address this explanation, just as you never commented on the extruded Koch snowflake with a bottom having the same "issue". It's no wonder you're trying to play the if-you're-wrong-that-means-I'm-right card.

To impose a limit on the infinite is to contradict. — Metaphysician Undercover

That sounds like a confused equivocation. A 1x1x1 cube by definition is 1 cubic units of volume, but it has an infinite number of points. It's not a contradiction to say that it has an unlimited number of points but a limited volume. Gabriel's horn has an unlimited extent into the x axis in the positive direction; that means it is unlimited... in extent... along the x axis. And that's it. It doesn't mean that the horn surrounds an infinite volume, as your equivocation is apparently meant to imply, any more than the fact that the 1x1x1 cube contains an infinite number of points suggests it should have an infinite volume.

The method is to apply a limit. — InPitzotl

Yes, apply a limit to what is stipulated by the premise, as without limit. That is the mistake. Can't you see that it is stipulated that there is no limit to the length of the horn, therefore to apply a limit is to contradict the premise?

That bag has all of the finite numbers in it, but no infinite numbers. — InPitzotl

You cannot put all the finite numbers in a bag, because there is an infinite quantity of them. This example provides nothing of relevance.

For example, a lower limit of 1 and an upper limit of 2 refers to "all of the numbers in the bag that are greater than or equal to 1, and less than or equal to 2". By contrast, a lower limit of 1 and an upper limit of ∞ simply means: "all of the numbers in the bag that are greater than or equal to 1". — InPitzotl

This is not relevant either. It is stipulated in the Gabriel's horn example, that there is no lower limit. It is stipulated that the horn continues infinitely. That means no limit. It is therefore not a case of having an upper limit and a lower limit, and to represent it as such is a mistake. To impose a lower limit (such as zero) is to contradict the premise of the example. We are not talking about how many numbers there are between two numbers here, we are talking about an unlimited length. To impose a limit on that length, a point where the diameter of the horn reaches zero, is to contradiction the premise of the example.

Wrong. The real issue is very simple... areas aren't volumes. "Paint" tricks you into thinking they are. It is very interesting to note that you never actually tried to address this explanation, just as you never commented on the extruded Koch snowflake with a bottom having the same "issue". It's no wonder you're trying to play the if-you're-wrong-that-means-I'm-right card. — InPitzotl

I did addressed this. If the horn can go infinitely thin, then so can the paint. They must play by the same rules.

That sounds like a confused equivocation. A 1x1x1 cube by definition is 1 cubic units of volume, but it has an infinite number of points. — InPitzotl

Again, we're not talking about an infinite number of points within a confined space, so that is irrelevant. The horn is infinitely long therefore there is no confined space. If someone were to say to you that there is an infinite extension of the universe, would you think that this implies a confined space? If someone says to you, take this line which continues infinitely, would you think that they were talking about a line segment which extends between two points?

This seems to be where your misunderstanding lies. You want to make this into an issue of a confined, "limited" space, but it is clearly stipulated that the horn is infinitely long, therefore there is no such confined space.

Surface area of a cylinder A = 2 * pi * r * h where r is the radius and h is the height.

Volume of a cylinder V = pi* r^2 * h where r is the radius and h is height

r approaches zero and h approaches infinity

A = 2 * pi * r * h = 2 * pi * (r approaching zero) * (h approaching infinity)

V = pi * (r approaching zero) * (r approaching zero) * (h approaching infinity)

(r approaching zero) * (h approaching infinity) = 1

So,

A = 2 * pi * 1 = 2 * pi

V = pi * (r approaching zero) * 1 = pi * (r approaching zero) = 0

As you can see, A plateaus to 2 * pi but V becomes 0.

V = pi * (r approaching zero) * (r approaching zero) * (h approaching infinity) , (r approaching zero) * (h approaching infinity) = 1 — TheMadFool

Suppose r=1/n and h=n^2. Then V -> pi. You are not describing Gabriel's Horn.

If infinity = z then, — TheMadFool

This is mysterious. One should make pronouncements about topics familiar to one.

(Not being a philosopher, this makes me wonder if some of the "sophisticated" philosophical arguments on the forum are any better) :roll:

Take it all with a grain of salt.

:lol:

Suppose r=1/n and h=n^2. Then V -> pi. You are not describing Gabriel's Horn. — jgill

We can't suppose anything we want. Gabriel's horn begins with the assumption that r = 1 and h extends from 0 to infinity.

Yes, apply a limit to what is stipulated by the premise, as without limit. That is the mistake. Can't you see that it is stipulated that there is no limit to the length of the horn, therefore to apply a limit is to contradict the premise? — Metaphysician Undercover

I can see that you're equivocating. You're confusing "limit" as a method with "limit" as a point beyond which you don't go.

You cannot put all the finite numbers in a bag, because there is an infinite quantity of them. — Metaphysician Undercover

The phrase "all the finite numbers" is itself such a bag.

This example provides nothing of relevance. — Metaphysician Undercover

Of course it's relevant. The 1/∞ that you're whining about is the 1/∞ in the video at 6:20. Right? That 1/∞ is 1/x with ∞ substituted in it. That 1/x is the 1/x from $\pi \frac{-1}{x}$. And that is from $\int_{1}^{\infty}{\pi \frac{1}{x^2}dx}$. And that is a Riemann integral; it's applied over the range where we want to take this volume. For Gabriel's horn, the lower limit here is 1, and there is no upper limit. To simply plug in infinity here as if it's a number is to say something different than what you yourself said in the prior post... that "infinite" simply means unlimited.

According to the rules of the method being applied, this is an improper integral. For improper integrals of this type, the calculation for the infinite portion is performed using a limit, whose definition I gave earlier.

It is stipulated in the Gabriel's horn example, that there is no lower limit. — Metaphysician Undercover

If there's no lower limit, then what's up with this big giant yellow arrow pointing to the lower limit during the segment where Gabriel's horn is defined?:
https://youtu.be/yZOi9HH5ueU?t=56
...not to mention Tom saying straight up, "and what we do first of all, is we're going to chop it here at 1"?

I did addressed this. If the horn can go infinitely thin, then so can the paint. They must play by the same rules. — Metaphysician Undercover

No, you didn't address this, because if this were indeed the case... if you could paint infinitely thin, then you can paint an infinite area with a finite amount of paint. And if you can do that, then there's no paradox. And you can do that, and there is no real paradox. But such infinitely thin paint simply becomes an empty metaphor... it's equivalent to what I was saying here. But you replied to that post saying that it wasn't the "real answer", and as recently as here you were peddling this one:

The "real answer" is that the horn is, stipulated by the stated premise, as infinitely long — Metaphysician Undercover

But your proposed "real answer" doesn't address the extruded Koch snowflake, because that isn't infinitely long. The thing you're proposing isn't the real answer does address the Koch snowflake, because the infinite area on its perimeter is not a volume.

Again, we're not talking about an infinite number of points within a confined space...This seems to be where your misunderstanding lies. You want to make this into an issue of a confined, "limited" space, but it is clearly stipulated that the horn is infinitely long, therefore there is no such confined space. — Metaphysician Undercover

You're reasoning by equivocation; "the extent is infinite, therefore the volume is infinite" simply doesn't follow.

The phrase "all the finite numbers" is itself such a bag. — InPitzotl

A phrase is a bag? Come on Pitzotl, you're reaching for straws. Get back to the subject.

No, you didn't address this, because if this were indeed the case... if you could paint infinitely thin, then you can paint an infinite area with a finite amount of paint. — InPitzotl

That's not true at all. Your interpretation of "infinite" is dreadful. You cannot paint an infinite area regardless of how much paint you have, because no matter how much painting you do there is always more to be painted. That's the issue with Gabriel's horn. It's infinitely long, so no matter how much paint you pour in the top, it never reaches the bottom.

It doesn't matter what you propose as the volume of the horn, you still cannot fill it with paint . Suppose you conclude it's 3.1 gallons,. You pour that in, but you haven't filled the horn because it hasn't reached the bottom.

But your proposed "real answer" doesn't address the extruded Koch snowflake, because that isn't infinitely long. The thing you're proposing isn't the real answer does address the Koch snowflake, because the infinite area on its perimeter is not a volume. — InPitzotl

We're discussing Gabriel's horn not snowflakes. How my answer relates to a snowflake is irrelevant. I don't see why you feel the need to bring up so many irrelevant issues.

You're reasoning by equivocation; "the extent is infinite, therefore the volume is infinite" simply doesn't follow. — InPitzotl

It seems you do not know the meaning of "equivocation".

A phrase is a bag? — Metaphysician Undercover

Yes.

Come on Pitzotl, you're reaching for straws. Get back to the subject. — Metaphysician Undercover

But the subject is the paradox of Gabriel's horn; it's literally the title of this thread. Gabriel's horn is an object defined using algebraic geometry. Algebraic geometry defines points in a space using coordinates using number lines. Number lines are defined with real numbers.

And that's what this bag is; it's the real numbers.

Ironically, your accusation and your imperative to me to get back to the subject is itself an avoidance of discussing this subject.

You cannot paint an infinite area regardless of how much paint you have, because no matter how much painting you do there is always more to be painted. ... Suppose you conclude it's 3.1 gallons,. You pour that in, but you haven't filled the horn because it hasn't reached the bottom. — Metaphysician Undercover

These are non-issues. In the paradox that is the subject of this thread, we're concerned primarily with the quantities not the length of the tasks. These are mathematical spaces; in the span of 30 seconds we define the entire infinite horn in an algebraic geometry space... doing infinite things is certainly not a problem here. Your imagined alleged problem has a lot more problems than you're letting on... the 3.1 gallons not going to the bottom is child's play. How are you upturning the horn in a gravitational field, and where do you put the planet? How does that planet manage to exert a field on the top of the horn anyway? But the biggest and most relevant question of all here is... do these really sound like mathematical questions?

We're discussing Gabriel's horn not snowflakes. How my answer relates to a snowflake is irrelevant. — Metaphysician Undercover

We're discussing the presumed paradox of Gabriel's horn. The presumptions that appear to introduce the conflicts is the point of the thread. Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite... the "outside-ness" of the surface of which is really an irrelevant detail that is part of the intuitive distraction of using paint to compare areas to volume. You are presuming that the paradox is solved by questioning the volume, but you haven't even shown a good reason to doubt the volume much less the flaw in the presumptions leading to the paradox.

Gabriel's horn is an object defined using algebraic geometry. Algebraic geometry defines points in a space using coordinates using number lines. Number lines are defined with real numbers. — InPitzotl

Nonsense. This has nothing to do with algebraic geometry. G's Horn is elementary calculus. :roll:

You guys should just let this go and get back to epistemological metaphysics where accuracy is optional.

This has nothing to do with algebraic geometry. — jgill

What do you mean this has nothing to do with algebraic geometry?:

Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of $x^2+y^2=1$ and is an algebraic variety, as are all of the conic sections. — Wolfram

Gabriel's horn is the algebraic variety defined by the polynomial z^2+y^2=(1/x)^2 starting at x=1.

G's Horn is elementary calculus. — jgill

Sure, you use calculus to analyze the surface area of and volume surrounded by this object, as they did in the video. But that doesn't preclude the fact that you're studying geometric properties of an algebraic variety.

These are non-issues. In the paradox that is the subject of this thread, we're concerned primarily with the quantities not the length of the tasks. — InPitzotl

This is where your mistaken. The paradox assumes a spatial form, created from mathematical principles, and names this form Gabriel's horn. Therefore the subject is this form which is created by mathematics, not the quantities which are used to create it. You need to divorce yourself from the numbers, from the mathematics, look at the form described, directly, and ask how is it to be measured.

This might be why you seem to be having so much difficulty understanding the problem. You want to use the same numbers which create the form, to measure the form. But when infinity is induced, this represents a failing in the capacity of the numbering system, so it is impossible to measure the form produced with the same numbers which produce it. That's what happens with the square root of two, and pi. There is a failure in the numbering system's capacity to measure the spatial form produced because of an inherent incommensurability. So we have a simple solution, we round off. Or, if necessary we can move toward employing more complex numbering systems, real numbers for example.

Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite. — InPitzotl

This is your false presumption, which is misleading you, that "the quantity of volume 'on the inside' is finite". If you would approach the problem with an open mind, rather than with what I see as a false presumption, we could probably make better progress in this discussion.

Here's what I propose. I'll justify my presumption, and you justify your presumption. I see that the spatial form which we are talking about, Gabriel's horn, is infinitely long. Therefore it is impossible, in theory, to precisely figure its volume. The volume therefore, in theory, is indefinite, being "infinite", unbounded, just like the extent of the natural numbers is "infinite", indefinite, or unbounded. We can however figure the volume of such forms, for practical purposes, by rounding off.

Will you justify your presumption that the volume is finite?

You are presuming that the paradox is solved by questioning the volume, but you haven't even shown a good reason to doubt the volume much less the flaw in the presumptions leading to the paradox. — InPitzotl

I told you already, Gabriel's horn is a spatial form with an infinite length. That's very good reason to doubt the volume. All you said was "You're reasoning by equivocation; 'the extent is infinite, therefore the volume is infinite'. Clearly there is no equivocation. A spatial form which has an unlimited (infinite) extension in one of its dimensions, will have an unlimited (infinite) volume accordingly

This is where your mistaken. — Metaphysician Undercover

Nothing you described justifies a concern about the length of tasks.

The paradox assumes a spatial form, created from mathematical principles, and names this form Gabriel's horn. — Metaphysician Undercover

The spatial form is given by Cartesian coordinates with three axes at right angles; that defines a space where the set of all points are (x, y, z) coordinates with x, y, z being reals.

Formatting for brevity.

"You want to use the same numbers which create the form, to measure the form." If it works.

"But when infinity is induced, this represents a failing in the capacity of the numbering system, so it is impossible to measure the form produced with the same numbers which produce it." That does not follow.

"That's what happens with the square root of two, and pi." The square root of two and pi are real numbers. See above. Also, square root of 2 is not infinite; it's between 1.4 and 1.5. Pi is not infinite; it's between 3.1 and 3.2.

This is your false presumption, which is misleading you, that "the quantity of volume 'on the inside' is finite". If you would approach the problem with an open mind, rather than with what I see as a false presumption, we could probably make better progress in this discussion. — Metaphysician Undercover

It's not a presumption; it's the result of a calculation. The fundamental issue here is that you're critiquing the methods without understanding what they are or why they are employed. Take your critique of Tom at 6:20 in the video for example. There, Tom is calculating the results of an integral. Tom has an improper integral, and this describes the method for its evaluation:

$\text{1. If }\lim_{a}^{t}{f(x)dx}\text{ exists for every }t \gt a\text{ then} \\ \hspace{2 em} \int_{a}^{\infty}{f(x)dx}=\lim_\limits{t \rightarrow \infty}{\int_{a}^{t}{f(x)dx}} \\ \text{provided the limit exists and is finite.}$ — Paul's online notes

When Tom says: "Well one over infinity, that's zero", that's okay, so long as we know it's a shortcut for:
$\hspace{2 em}\lim_\limits{x \rightarrow \infty}{\frac{1}{x} }$
...which is exactly 0, as shown previously by definition of that limit. You have confused this with saying that 1/infinity=0. That's baseless; infinity is not a real number; the domain of the integral is the same domain as the x axis, and infinity isn't even in that domain. The method isn't "plug in infinity", and there's a reason it isn't.

I see that the spatial form which we are talking about, Gabriel's horn, is infinitely long. Therefore it is impossible, in theory, to precisely figure its volume. — Metaphysician Undercover

That does not follow.

Will you justify your presumption that the volume is finite? — Metaphysician Undercover

That was already shown, and you're mischaracterizing the problem. The most fundamental problem here is that you're objecting to the efficacy of these methods without understanding what the methods are being employed or why they are employed. The other big problem is the obvious bias portrayed in objecting to the efficacy of the method before understanding these things.

A spatial form which has an unlimited (infinite) extension in one of its dimensions, will have an unlimited (infinite) volume accordingly — Metaphysician Undercover

That does not follow.

It's not a presumption; it's the result of a calculation. The fundamental issue here is that you're critiquing the methods without understanding what they are or why they are employed. Take your critique of Tom at 6:20 in the video for example. There, Tom is calculating the results of an integral. Tom has an improper integral, and this describes the method for its evaluation:
1. If limtaf(x)dx exists for every t>a then∫∞af(x)dx=limt→∞∫taf(x)dxprovided the limit exists and is finite.1. If limatf(x)dx exists for every t>a then∫a∞f(x)dx=limt→∞∫atf(x)dxprovided the limit exists and is finite.
— Paul's online notes
When Tom says: "Well one over infinity, that's zero", that's okay, so long as we know it's a shortcut for:
limx→∞1xlimx→∞1x
...which is exactly 0, as shown previously by definition of that limit. You have confused this with saying that 1/infinity=0. That's baseless; infinity is not a real number; the domain of the integral is the same domain as the x axis, and infinity isn't even in that domain. The method isn't "plug in infinity", and there's a reason it isn't. — InPitzotl

Well you distinctly said it was a presumption. "Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite..."

We've been through all this. Your so-called calculation, "limx→∞1xlimx→∞1x...which is exactly zero by definition..." is nothing but a rounding off. See, it's zero by definition, not by calculation.

What do you mean this has nothing to do with algebraic geometry? — InPitzotl

Whereas one can describe the collection of points in 3-space comprising GH with the zeros of
$P\left( x,y,z \right)={{z}^{2}}{{x}^{2}}+{{y}^{2}}{{x}^{2}}-1$, the paradox of GH does not emanate from that perspective, but from elementary calculus. Why even bring varieties up since it is irrelevant to the issue being discussed, and participants of the thread might well be familiar with the rudiments of calculus, but have little acquaintance with algebraic geometry?

That does not follow. — InPitzotl

I see that we have a fundamental difference of opinion concerning the logic of spatial areas. I think that it is illogical to believe that a 3d spatial form with an infinite extension in one dimension could have a finite volume. You disagree.

I see that we have a fundamental difference of opinion concerning the logic of spatial areas. I think that it is illogical to believe that a 3d spatial form with an infinite extension in one dimension could have a finite volume. — Metaphysician Undercover

Well ok. What would you say is the volume of the solid of revolution of $y = \frac{1}{x}$ between 1 and $\infty$ when the curve is revolved around the x-axis? Here's the theory and formula, if you forgot.

https://en.wikipedia.org/wiki/Solid_of_revolution