Thought Experiments in Mathematics: Gabriel's Horn
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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.
Of course, this being mathematics, changing the rules can produce a counterexample. A simple way of doing that is by declaring the inside of the shape the outside and vice versa.
I expect one could construct a counterexample around a black hole, too.
And this idea of finite area described by y = f(x) [between this curve and the x-axis, say], with infinite perimeter, amounts to just the same thing some function f whose integral over an infinite range (e.g., from x = 1 to infinity) is finite. If you prefer to think discretely, this is essentially the same phenomenon as an infinite series with finite sum (splitting our curve into the block from x = 1 to 2, the block from x = 2 to 3, the block from x = 3 to 4, etc., the total area is the sum of the series of areas in these blocks, while the total perimeter is automatically infinite as at least length 1 on top and bottom is contributed in each of these blocks). So, consider, for example, y = 1/x^2, from x = 1 to infinity: infinite perimeter but finite area (as its antiderivative is -1/x + C, which only increases by a finite amount over this range; in the same way, differentiating any function with a finite asymptote yields examples, and every example comes in essentially this fashion).
In moving to 3d and making a surface of revolution out of our starting curve, we are now integrating πf^2 instead of f itself, so we need f^2 to yield a convergent integral rather than f itself, but otherwise everything is just the same. Thus, we can get away with f(x) = 1/x, as in Gabriel's Horn, but also, just as well, with f(x) = x^{-p} for any p > 1/2.
Not that these are the only such things; again, any convergent infinite integral or series yields examples. So, just as well, y = 1/2^x or such things would work.
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"
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....
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 :-)
I dont know that it would be exactly analogous to normal Z[M+1], and might have useful properties for some kind of geometric or combinatorical modeling. (My hunch is any time you want to be capable of carrying a "crossed threshold" flag as well as a value, and have that cascade through calculation.)
It would also model systems where you can get to infinity in finite steps, but can't traverse back. Not sure if those are useful in the abstract, though.
That's pragmatic, highly useful, but a nightmare for mathematicians. You lose invariants such as x+y-y=x (associativity and communicativity, in general), so you're no longer talking of a group (I don't know of research on 'almost groups')
Compiler writers may happily make matters worse by assuming those laws still hold, with the effect that the same computation may overflow or not on different CPUs, under different compilers, compiler settings, or even the same compiler in the same compilation run.
Well, you actually just lose general inverses, so there's no sensible '-' operation that can just be turned in to '+ (-x)', where -x is the inverse of x under +. It's similar to the case where you have x * y / y = x except in the case where y = 0 (because there's no 0^-1). (As someone else pointed out, this puts you in to a semi-ring rather than a ring, which is what the general tropical semi-ring is. [1])
You can still have elements for which x + y - y = x is true, and usually that's a well-defined sub-space of x,y combinations. Proofs then usually use a case analysis: either we're in the subspace where x + y - y = x holds, or else we can use a property of not being in that space to derive a different but still useful conclusion.
In some senses, it acts as a Maybe type, and can perform arithmetic without having to unpack that Maybe-ness.
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.
That the horn's tail (or mouth piece, as it were) extends into an infinitely small value approaching zero, is as fair as the never-ending irrational fraction pi represents.
Otherwise, the horn's volume is undefined. Where does the tip of the horn cross zero?
It doesn't??? Hmmm, that sounds like a non-finite value to me.
Discussions like this often wax political more than analytical. There are four lights.
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.
To say that pi is a non-repeating fraction with infinite decimal places, is to say that we cannot ever precisely know the boundary and definition of a circle, forever zooming in on a point never met.
That we find circles, and capitulate to what must be their true or approximate location is a cheat, and mere luck at the integrity of atoms or planck space or what have you.
But still, it's garbage in the same way that the divide-by-zero rule is garbage. Granted, dividing something into nothing isn't a practical operation. But, I can divide zero into as many parts as I like, and it's still zero.
So, if people want to keep stacking infinitesimally small tenths, hundreths, thounsandths, millionths and so on, to the end of any particular number, forever, well, forever sounds like an infinite quantity to me, and I'll argue to point that out.
Unroll the circumference of a circle. Where does it end? Approximately near X? Approximately???
Tell me the area of a circle. Approximately within this fuzzy boundary? Approximately???
I am dubious.
That may be. Or it might not. Whether (physical) space is infinitely divisible or not is, I think, an open question. I think it is even plausible that the answer might be that it can be described equally well both ways? Not sure.
I don't mind thinking about abstract objects which do not physically exist. (Though, I generally consider abstract objects to exist in some sense, though you may disagree, and that's fine.)
If you don't care to think about them, that's fine also.
I believe that thinking about them is often useful. Treating pi as existing allows for reasoning about things more efficiently, even if it turns out that it results in inaccuracies in predictions about the physical world which are far smaller than our ability to measure/notice. (If we can't notice the results of the error, it doesn't seem to be much of a problem.)
That doesn't mean you have to accept pi as actually existing. I think it would be to your benefit to at least accept its use though. I do consider it to actually exist.