> On Tuesday, July 22, 2014 7:57:07 AM UTC-4, muec...@rz.fh-augsburg.de wrote: >> > For example, we all agree that a line has 0 area. >> > Take a rectangle whose dimensions are r by 1/r. >> > The area stays r*1/r = 1 through out the whole process. >> > The rectangle, in the limit, approaches a line segment. >> > Now, you don't see WM complaining about area >> >> No. There is no actual infinity. Why should it exist in geometry? > > Dear Students of WM, > > Here, we see WM claim that infinity does not exist in geometry. So > lines, spirals, space-filling curves, all have finite length. Even > unbounded regions have finite area (or volume). > > So according to WM, the length of a line in the xy-plane should be a > finite number.
No, that's not what he says! (I realise how odd it is my coming in like this, but I think I have gained some knowledge of exactly what WM's nonsense really says.)
He will not say that an unbounded line has a finite length. His claim is that there is no *actual* infinity. All the magic is in that one word. Similarly he won't claim that there is a largest natural, but that the set is "never completed". You have to remember that it's all about words. None of these thing will ever be defined, but it makes for a good story to tell students confronted with the peculiarities of infinite sets. The marketing line "'every' is not 'all'" is particularly beguiling.
The trick (and it's not a bad one) is to avoid ever saying what these very pleasantly suggestive words mean in any formal sense. When you press him, he claims that certain things follow from having actual or completed infinities, but nothing even remotely like a proof is ever produced.
> Here's the example again. > > R_n = n by 1/n rectangle. If you fix one corner of the rectangle and > make n->oo, this geometric object turns into a 1-dimensional ray. > > Area(R_n) = 1 > > Ask your professor what > > lim n-> oo Area(R_n) = 1 means.
It means that, for every finite or "given" n, the area is n * 1/n = 1. I don't see how the limit would trouble him. I do see your point, but since he does not think that unbounded lines are finite, there's no problem. He thinks, remember, that all *actual* lines are finite.
Limits work without whatever it is "completed" infinities are. Hence the illogicality of his claim that limit sets reply on completed infinities since they can be defined in terms of plain numerical limits. (It's possible he has not bothered to read up on the indicator function definition, or he may just be deliberately ignoring it.)
To me, there is only one question that really matters, and that is what axioms, definitions, or rules of logic must change in order to exclude completed or actual infinities, and this avoid the supposed "contradictions". If we know, we could all tell what theorems remain and which get lost, but you can be certain that this question will never be answered by him. Well, it will, of course, because WM can answer *any* question if he uses enough words to obscure all the technical details. I mean that he won't answer it in a way that would let you or I determine what theorems remain valid. He must always remain the only person who can tell.