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Thought Experiments in Mathematics: Gabriel's Horn

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Re: Thought Experiments in Mathematics: Gabriel's Horn

#11

Post by llamaz »

Someone wrote: post #9
zeroer wrote: post #4

The most interesting thing about it is that at one point people thought it was a paradox.

It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers." . I even fear that is true for the easier to believe "0.9999999... = 1"

There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers.

For example, most mathematicians would agree that there are as many reals in [0, 1] as there are in the real line. But according to a different definition of size, namely the most famous measure from measure theory, the Lebesgue measure --- the length of [0, 1] is 1 while the real line is undefined (I think). The person on the street would probably prefer the Lebesgue measure as more intuitive.

I guess my ultimate point is that imprecise statements are harder to prove wrong. There's a quote, I don't remember who said it, that you should mistrust precise statements rather than imprecise ones, because it's precisely precise statements that can be proven wrong.

edit: found the quote: http://www.brainyquote.com/quotes/quotes/r/raymondsmu190329....

Re: Thought Experiments in Mathematics: Gabriel's Horn

#13

Post by tanderson92 »

llamaz wrote: post #11
Someone wrote: post #9

Earlier quoted context omitted.

It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers." . I even fear that is true for the easier to believe "0.9999999... = 1"

There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers. For example, most mathematicians would agree that there are as many reals in [0, 1] as there are in the real line. But according to a different definition of size, namely the most famous measure from measure theory, the Lebesgue measure --- the length of [0, 1] is 1 while the real line is undefined (I th…

But the Lebesgue measure is not a counting measure so it cannot be in any way related to the "number of numbers"...?

Re: Thought Experiments in Mathematics: Gabriel's Horn

#14

Post by danbruc »

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.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#15

Post by Someone »

llamaz wrote: post #11
Someone wrote: post #9

Earlier quoted context omitted.

It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers." . I even fear that is true for the easier to believe "0.9999999... = 1"

There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers. For example, most mathematicians would agree that there are as many reals in [0, 1] as there are in the real line. But according to a different definition of size, namely the most famous measure from measure theory, the Lebesgue measure --- the length of [0, 1] is 1 while the real line is undefined (I th…

"There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers."

You could call Z[n] ?for any composite n? that (in Z[4], the multiplication table only contains 0, 1, and 2, so 3 is prime there; Z[p] for prime p gives you p different numbers and zero primes) but I think those are the only ones. If you accept that infinity exists you get Hilbert's hotel, which gets you all those paradoxes, which after lots of sleepless nights leads to the only logical conclusion that giving up intuition about infinities is the best way out.

If you don't accept that infinities exist, there must be a largest integer M, and you get to decide what M+1 or 2M are. That leads either to Z[n], to K&R's undefined behavior, which is so ugly no mathematician would dare publish it :-), or to some formalized variant of it that isn't Z[n].

I'm not sure I would call the values of Z[n] natural numbers, though, as that feels like it requires having negative numbers, too. Hm, maybe a shifted Z[n] would work. If you replace {0,1,2,3} by {0,1,2,-2,-1} in Z[5], you have two negative and three natural numbers in your universe, none of which is prime.

I doubt that any of this kind of mathematical hair-splitting would bring aboard those who have trouble with grasping 0.999999... = 1, though :-)

Re: Thought Experiments in Mathematics: Gabriel's Horn

#16

Post by msds »

llamaz wrote: post #11
Someone wrote: post #9

Earlier quoted context omitted.

It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers." . I even fear that is true for the easier to believe "0.9999999... = 1"

There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers. For example, most mathematicians would agree that there are as many reals in [0, 1] as there are in the real line. But according to a different definition of size, namely the most famous measure from measure theory, the Lebesgue measure --- the length of [0, 1] is 1 while the real line is undefined (I th…

The real line is defined, as Lebesgue measure spits out the extended reals.

Re: Thought Experiments in Mathematics: Gabriel's Horn

#17

Post by [deleted] »

llamaz wrote: post #11
Someone wrote: post #9

Earlier quoted context omitted.

It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers." . I even fear that is true for the easier to believe "0.9999999... = 1"

There probably is _some_ mathematical formalism out there where there are less prime numbers than natural numbers. For example, most mathematicians would agree that there are as many reals in [0, 1] as there are in the real line. But according to a different definition of size, namely the most famous measure from measure theory, the Lebesgue measure --- the length of [0, 1] is 1 while the real line is undefined (I th…

[deleted]

Re: Thought Experiments in Mathematics: Gabriel's Horn

#18

Post by aoki »

Jun8 wrote: post #2

This answer on Math SE illustrates the concept of finite volume-infinite area quite nicely ( http://math.stackexchange.com/a/14632/14643 ). Note that the converse is proven only for surfaces of revolution and for differentiable functions. Finding a pathological counter example that violates these assumptions would be interesting.

ah, thank you. i did not really understand why this was really any more "surprising" than the 2D analogue (finite area under an infinite curve). there are at least some other people who think it's not :-)

Re: Thought Experiments in Mathematics: Gabriel's Horn

#19

Post by tgb »

Chinjut wrote: post #6
Jun8 wrote: post #2

This answer on Math SE illustrates the concept of finite volume-infinite area quite nicely ( http://math.stackexchange.com/a/14632/14643 ). Note that the converse is proven only for surfaces of revolution and for differentiable functions. Finding a pathological counter example that violates these assumptions would be interesting.

The converse is impossible in any context where the isoperimetric inequality is applicable.

Good point. Though the Wikipedia article on it gives the most general form as requiring our region to have closure with finite Lebesgue measure. This is necessary of course by the example another corner brought up about swapping the inside and outside. But that means it exactly can't rule out the existence of regions with infinite volume (outside and inside) and finite surface area!

Re: Thought Experiments in Mathematics: Gabriel's Horn

#20

Post by amelius »

Someone wrote: post #9
zeroer wrote: post #4

The most interesting thing about it is that at one point people thought it was a paradox.

It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers." . I even fear that is true for the easier to believe "0.9999999... = 1"

It is because using infinity in place of a number is really a hack. And it is funny to see that mathematicians also get confused by their own hacks :)
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