Cantor Confusion
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From: David Marcus on 10 Oct 2006 14:46 Virgil wrote: > In article <J6w6LC.9rL(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: > > > In article <virgil-9E9CC6.02103209102006(a)comcast.dca.giganews.com> Virgil > > <virgil(a)comcast.net> writes: > > > > Dik T. Winter wrote: > > ... > > > > > The balls in vase problem suffers because the problem is not > > > > > well-defined. Most people in the discussion assume some implicit > > > > > definitions, well that does not work as other people assume other > > > > > definitions. How do you *define* the number of balls at noon? > > ... > > > How about the following model: > > > > And you also start with definitions, or a model. I did *not* state that > > it was difficult to define, or to make a model. But without such a > > definition or model we are in limbo. I think other (consistent) definitions > > or models are possible, giving a different outcome. > > Can you suggest one? One that does not ignore the numbering on the balls > as some others have tried to do. Models that ignore the number of the balls are certainly models. Whether they are reasonable translations of the original problem into Mathematics is a separate question. One might naturally ask how such translations would be done if the problem was changed slightly, e.g., by changing the number of balls or the times. -- David Marcus
From: Albrecht on 10 Oct 2006 15:34 William Hughes wrote: > Albrecht wrote: > > William Hughes schrieb: > > > > > Albrecht wrote: > > > > William Hughes schrieb: > > > > > > > > > Albrecht wrote: > > > > > > > > > > <...> > > > > > > > > > > > I don't controvert the axiomatic methode anymore. But I claim that it > > > > > > isn't the only and the important one in math. In teaching and in the > > > > > > mind of the people the axiomatic method appears to be the only right > > > > > > way to do math. That's not correct. > > > > > > The nondenumerable infinity of the reals is not the only one truth. > > > > > > Nobody is wrong who claims only one kind of infinity, the one we only > > > > > > can know: the endless infinity. > > > > > > > > > > > > > > > > > > > > > The problem is not that someone who believes > > > > > in your intuitive "endless infinity" (intuitive because it cannot > > > > > be put on a mathematical footing) is wrong. > > > > > > > > Oh yes, it is the problem. I came to these subject by reading a bunch > > > > of popular books about math. When I read the diagonal argument the > > > > third or fourth time I started to wonder. The textes were of differnt > > > > quality but all of them had a special sort of feeling. And all stated, > > > > that this proof is so elementary, easy and absolute right that nobody > > > > had anything to reflect or critizise about it. > > > > But I found in shortest time a lot of questions about the issue. > > > > Later I read professional works about set theorie and I found a similar > > > > feeling in the textes about the diagonal argument. And then I started > > > > to learn about the role of ZF in the teaching on universities and I had > > > > a lot of disputes about the matter in newsgroups. > > > > > > Don't you find it interesting that of all the places you looked, > > > the only place where anyone disagreed with the diagonal > > > argument was the newsgroups? > > > > That's really untrue. I had read several books and papers of academics > > (who do not post in this or the german math newsgroups) in which they > > formulate (very cautious) criticism about ZF, axiomatic set theory or > > especially the axiom of infinity. > > Did any of them disagree with the diagonal argument? I think, the answer to this question don't enlight the whole problem we discuss. Does someone agree or disagree with the diagonal argument who reject the existence of infinite objects like infinite sets? What do you think? Just a short seach in my papers gave this: http://web.maths.unsw.edu.au/~norman/views.htm (see the discussion a short time ago in sci.logic) Prof. Rudolf Taschner, Institut für Analysis und Scientific Computing, Technische Universität Wien Titel: Das Unendliche, Berlin 2006 (2. Auflage) Pierre Basieux, author Titel: Abenteuer Mathematik, Reinbeck bei Hamburg 2002 (3. Auflage) .... Best regards Albrecht S. Storz > > This is the problem. Not someone saying, "your arguments lead > to conclusions I find counterintuitive, so the axioms you chose need > to be modified", but someone saying "your arguments lead > to conclusions I find counterintuitive, so there must be something > wrong with your arguments" or "your arguments lead > to conclusions I find counterintuitive, so your axioms must > be inconsistent". > > - William Hughes
From: David R Tribble on 10 Oct 2006 16:08 Mueckenheim wrote: >> But you cannot derive that the vase is not empty at noon from the >> observation that its contents cannot decrease? > Han de Bruijn wrote: >> A picture says more than a thousand words. [Doesn't] it? >> >> http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > David R Tribble schreef: >> I notice that there is no Y point at the rightmost X at "noon". > Han.deBruijn wrote: > True. That symbolizes the fact that there is no noon. > It's also a fact that you cannot do something else with infinity > than clipping it against the window, graphically speaking. I get it now. We can never "get to" noon, so it does not exist. Likewise, the infinite sum s = 1/2 + 1/4 + 1/8 + 1/16 + ... can never be "reached", so it is not actually equal to 1.
From: David R Tribble on 10 Oct 2006 16:27 Dik T. Winter wrote: >> And you also start with definitions, or a model. I did *not* state that >> it was difficult to define, or to make a model. But without such a >> definition or model we are in limbo. I think other (consistent) definitions >> or models are possible, giving a different outcome. > Virgil wrote: >> Can you suggest one? One that does not ignore the numbering on the balls >> as some others have tried to do. > David Marcus wrote: > Models that ignore the number of the balls are certainly models. Whether > they are reasonable translations of the original problem into > Mathematics is a separate question. I tried modeling the problem with sets: B_0 = { } B_1 = B_0 U {1,2,3,...,10} \ {1} ... B_n = B_(n-1) U {10(n-1)+1,10(n-1)+2,...,10(n-1)+10} \ {n} ... The resulting "balls in the vase at noon" would then seem to be B_w. Which is problematic, since there is no ball labeled "w". So this is probably a bad model of the problem. (I don't claim to be an expert.) > One might naturally ask how such > translations would be done if the problem was changed slightly, e.g., by > changing the number of balls or the times. I always thought the problem would be simpler if, at each step, two balls were added to the vase and then one was removed. Same result, of course, but the equations are a bit shorter.
From: Randy Poe on 10 Oct 2006 16:31
David R Tribble wrote: > Mueckenheim wrote: > >> But you cannot derive that the vase is not empty at noon from the > >> observation that its contents cannot decrease? > > > > Han de Bruijn wrote: > >> A picture says more than a thousand words. [Doesn't] it? > >> > >> http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > > > > David R Tribble schreef: > >> I notice that there is no Y point at the rightmost X at "noon". > > > > Han.deBruijn wrote: > > True. That symbolizes the fact that there is no noon. > > It's also a fact that you cannot do something else with infinity > > than clipping it against the window, graphically speaking. > > I get it now. We can never "get to" noon, so it does not exist. > > Likewise, the infinite sum > s = 1/2 + 1/4 + 1/8 + 1/16 + ... > can never be "reached", so it is not actually equal to 1. Worse than that, when you write that expression, 1 ceases to exist. - Randy |