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Thought Experiments in Mathematics: Gabriel's Horn

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Re: Thought Experiments in Mathematics: Gabriel's Horn

#21

Post by ktta »

danbruc wrote: post #14
mkagenius wrote: post #12

How is this different from this: "a plain sheet of paper can have infinite surface area and finite (0) volume" ?

An ideal sheet of paper, a plain, is a two dimensional object and has by definition no volume. In case of Gabriel's horn we really have a three dimensional object bounded by a two dimensional surface. The bounding surface itself has no volume just like a plain.

But an ideal sheet of paper can have finite surface area and infinite perimeter length. See the Stackoverflow link in a comment above.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#22

Post by danbruc »

ktta wrote: post #21
danbruc wrote: post #14

Earlier quoted context omitted.

An ideal sheet of paper, a plain, is a two dimensional object and has by definition no volume. In case of Gabriel's horn we really have a three dimensional object bounded by a two dimensional surface. The bounding surface itself has no volume just like a plain.

But an ideal sheet of paper can have finite surface area and infinite perimeter length. See the Stackoverflow link in a comment above.

Of course, in both cases it is interior - volume respectively area - and boundary - surface area respectively boundary length - but that is not the same thing as area and volume of a two dimensional object. The analog for Gabriel's horn would be volume and four dimensional hypervolume which is of course also zero because it is a three dimensional object. The paradox goes from interior to boundary, what mkagenius suggested is taking the interior one dimension up.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#23

Post by ezequiel-garzon »

danbruc wrote: post #14
mkagenius wrote: post #12

How is this different from this: "a plain sheet of paper can have infinite surface area and finite (0) volume" ?

An ideal sheet of paper, a plain, is a two dimensional object and has by definition no volume. In case of Gabriel's horn we really have a three dimensional object bounded by a two dimensional surface. The bounding surface itself has no volume just like a plain.

I cannot resist: it's a plane, not a plain.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#24

Post by blippityBlop »

Okay, tra la la. But can we all take note that the volume is PI, and that PI's decimal places are also infinite.

In this moment, I take issue with the idea that PI represents a "finite" volume.

PI, is more accurately a "known-volume-other-than-infinity."

Meanwhile, angels dance upon the head of a pin somewhere.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#25

Post by tedunangst »

blippityBlop wrote: post #24

Okay, tra la la. But can we all take note that the volume is PI, and that PI's decimal places are also infinite. In this moment, I take issue with the idea that PI represents a "finite" volume. PI, is more accurately a "known-volume-other-than-infinity." Meanwhile, angels dance upon the head of a pin somewhere.

uhm, 2/7 has infinite decimals. it's not infinite.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#26

Post by danbruc »

ezequiel-garzon wrote: post #23
danbruc wrote: post #14

Earlier quoted context omitted.

An ideal sheet of paper, a plain, is a two dimensional object and has by definition no volume. In case of Gabriel's horn we really have a three dimensional object bounded by a two dimensional surface. The bounding surface itself has no volume just like a plain.

I cannot resist: it's a plane, not a plain.

Damn it, I even looked it up because I was unsure. But the German word Ebene translates to plane and plain and I missed that.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#27

Post by drdeca »

blippityBlop wrote: post #24

Okay, tra la la. But can we all take note that the volume is PI, and that PI's decimal places are also infinite. In this moment, I take issue with the idea that PI represents a "finite" volume. PI, is more accurately a "known-volume-other-than-infinity." Meanwhile, angels dance upon the head of a pin somewhere.

You can just re-scale it to have volume 1 if you want.

Also, uh, you realize that pi is less than 5, and more than 0, right?

If you draw a circle with radius 1, the area would be pi just as much as the volume of this at some scaling.

Would you say that such a circle has infinite area? That seems like it would be refusing to use a term as it is generally used. You can object by saying that circles do not exist, I suppose. That would be consistent.

But saying that a unit circle has infinite area is either false, or using a term in a nonstandard way.

Similarly, you could reject such a horn (perhaps because it isn't bounded), but calling the volume infinite is like calling the area of a unit circle infinite.

Maybe you just object to abstract objects in general?

If so, alright. People are still going to study them though.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#30

Post by [deleted] »

ezequiel-garzon wrote: post #23
danbruc wrote: post #14

Earlier quoted context omitted.

An ideal sheet of paper, a plain, is a two dimensional object and has by definition no volume. In case of Gabriel's horn we really have a three dimensional object bounded by a two dimensional surface. The bounding surface itself has no volume just like a plain.

I cannot resist: it's a plane, not a plain.

[deleted]
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