Cantor Confusion
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From: Han de Bruijn on 22 Nov 2006 06:08 Eckard Blumschein wrote: > On 11/22/2006 9:22 AM, Han de Bruijn wrote: > >>Eckard Blumschein wrote: >> >>>To what extent criticism by WM might be justified? >>>Let me tell you my position: WM is most likely wrong if he sees the >>>finished infinities the fundament of modern mathematics. >> >>Quite on the contrary. >> >>Eckard, please learn more about Wolfgang Mueckenheim in the first place: >> >>http://www.fh-augsburg.de/~mueckenh/ >> >>Wild guess: his ideas will turn out to be not so uncomfortable with you. > > Hallo Han, > > First of all I would like to urge you for restricting the discussion to > sci.mat by removal of de.sci.mathematik. Sorry. I didn't notice this crosspost. > Nice to meet you again. I am also glad that you are trying to defend WM. > I benefitted a lot from his booklet "Die Geschichte des Unendlichen" > which was sent to me by him personally. Okay. That's better. > Presumably you got me wrong concerning what I consider the fundament of > modern mathematics. I consider set theory a toxic coat of modern > mathematics rather than its genuine fundament. Modern mathematics is > inconceivable without the contributions e.g. by Thales, Eudoxos, > Pythagoras, Archimedes, Euclid, Fermat, Vieta, Descartes, Galilei, > Newton, Leibniz, Euler, Bernoulli, Laplace, Lagrange, Fourier, Hermite, > Lindemann, Gauss, Cauchy, Kronecker, Poincar?, Shannon, Kolgomoroff, > and even Bill Gates. Dedekind and others contributed an illusion > necessarily leading to endless quarrels. Don't forget Chebyshev, an excellent mathematician I've discovered quite recently. His polynomials turn out to be essential for some recent work of mine. Maybe read the last paragraph in: http://hdebruijn.soo.dto.tudelft.nl/jaar2006/drievoud.pdf > WM has to adapt too much to present mathematical terminology. When he > confused Paul with Emile this was an excusable mistake to me. > However I cannot follow him completely. Han de Bruijn
From: Han de Bruijn on 22 Nov 2006 06:13 Eckard Blumschein wrote: > Presumably you got me wrong concerning what I consider the fundament of > modern mathematics. I consider set theory a toxic coat of modern > mathematics rather than its genuine fundament. [ ... snip ... ] It could be that we agree completely on this: http://groups.google.nl/group/sci.math/msg/9c2f02ed63157fcc?hl=nl& Han de Bruijn
From: mueckenh on 22 Nov 2006 06:43 Virgil schrieb: > > One counter example contradicts ZFC: > > There is not one single element of the diagonal which is not contained > > in a line. This line contains this and all preceding elements. > > While true, it is irrelevant to the issue of whether there is one line > containing every element of the diagonal, which there is not. > > WM conflates "every element of the diagonal is in SOME line", which is > true, with "every element in the diagonal is in THE SAME line", which > is false. Tha means we need at least two lines for the elements o the diagonal? Please give an example which requires that at least two different lines are needed to contain two elements of the diagonal. Regards, WM
From: mueckenh on 22 Nov 2006 06:45 Virgil schrieb: > In article <456304B0.70705(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > Being admittedly not very familiar with set theory, I nonetheless wonder > > if sets are considered like something going on forever. > > Sequences do, sets do not. > Sequences are sets. Regards, WM
From: mueckenh on 22 Nov 2006 06:48
Franziska Neugebauer schrieb: > You have not commented on the first part of my posting: No. > >> Do you want to posit, that there is some line identical to the > >> diagonal? That is not (yet) proven. > > > > The number of elements of the diagonal is larger than all n: > > E omega A n : omega > n. > > 1. n is not suitably constrained. For any n > omega the formula > is omega > n is wrong hence the formula is wrong. Please read yourself. In future I am not willing to explain everything twice to you. We discuss a diagonal with omega natural elements. > > > The lines have only finitely many elements. > > > > In a linear order of finitely many elements n exactly the following is > > implied: > > If for every n there exists an line L(n) such that L(n) contains all > > elements m <=n > > then there exists a line L such that for every n, L contains all > > elements m <=n . > > > This is valid for the finitely many elements of every line. (How many > > lines there are is irrelevant.) > > Proof? It is a self evident truth for finitely many elements like the elements of a line, because every finite set has a largest element. There exists no counter example, and it is impossible to construct a counter example. Regards, WM |