Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Albrecht on 5 Oct 2006 03:28 MoeBlee schrieb: > Albrecht wrote: > > MoeBlee schrieb: > > > > > Albrecht wrote: > > > > > > > I will try to explain my claim again: There is only a[n] [anti-]diagonal number > > > > which is proveable not in the list if there is a real number which is > > > > build up out of all the real numbers in the list. But for an infinite > > > > list you can't end the diagonal number. You may have a sequence which > > > > converge. But you never have a limit. > > > > > > Then you're not talking about the reals. > > > > The points on a straight line are representations of the reals. Do you > > think so? > > No, I don't think they are "representations" of real numbers. In > analytic geometry, in, for example, the plane R2, they are ordered > pairs of real numbers. Now let one of the numbers of the pairs constant and forget it and we meet. > > > With Cantor you have to think that some points on the > > straight line (to be exact: the absolute majority of them) are not > > reachable by construction. > > Should this be a consequence of an useful mathematics? > > Give me your axiomatization that is rich enough for ordinary calculus, > and with a mathematical definition of 'constructible', and such that > every real number is constructible. Then we'll talk about that. Do we really need such an axiomatisation? 99% of mathematics was done without it. And was done right. > > > Are the consequences you claim for the Cantor proof really needful? I > > don't think so. > > Whether needed or not, given an axiomatization, they're theorems. > > > There are many examples in which the systemes break down in > > extrapolation to infinity. Let's have the sequence of polygones in the > > circle which numbers of edges run up to infinity. Extrapolate the areas > > to the limite case and you will get the area of the circle. Isn't it > > rational to think that the the circle is identic with the limit case of > > the polygones? Isn't it rational to think that the limit case of the > > polygones is a polygone with (countable) infinite edges and this edges > > are identical with (all) the points on the circle line? > > Since we don't even agree as to what a real number is, I can't imagine > a discussion with you about the above. > > > Concerning Cantors diagonal argument: Can we rational speak about > > convergence if we don't have a (finite) law which gives us endlessly > > the converging values? > > I don't know what you mean by "endlessly converging values". What is > the difference between converging and "endlessly converging"? > Surely my explanation was not very well done. But I feel that you don't want more explanations about my standpoint anyway. 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. Best regards Albrecht S. Storz
From: Han de Bruijn on 5 Oct 2006 03:42 David R Tribble wrote: > Albrecht wrote: > >>>The idea of "all infinite many objects" in any sense is >>>self-contradicting. There is no "all" of infinite many things. > > Han de Bruijn wrote: > >>Bravo! Another mathematician (?) who cares about disciplinary thinking. > > Let's see. If I define set N as containing "all" of the naturals, i.e., > {0,1,2,3,...}, you're (both) saying that N can't exist as a real set, > because there is no way to have "all" of an "infinitely many" elements > at once. > > So then we must conclude that there are naturals that are not in N, > because N can't contain all the (infinitely many) naturals. > > So now I define another set N' to contain all the naturals except those > that are already in N, i.e., those naturals that we concluded must be > omitted from N. > > So what is in this second set N'? Is it also not an infinite set? You have just demonstrated that infinitary set theory does not work. Han de Bruijn
From: Virgil on 5 Oct 2006 03:49 In article <1160033333.428361.122020(a)h48g2000cwc.googlegroups.com>, "Albrecht" <albstorz(a)gmx.de> wrote: > MoeBlee schrieb: > > > Albrecht wrote: > > > MoeBlee schrieb: > > > > > > > Albrecht wrote: > > > > > > > > > I will try to explain my claim again: There is only a[n] > > > > > [anti-]diagonal number > > > > > which is proveable not in the list if there is a real number which is > > > > > build up out of all the real numbers in the list. But for an infinite > > > > > list you can't end the diagonal number. You may have a sequence which > > > > > converge. But you never have a limit. > > > > > > > > Then you're not talking about the reals. > > > > > > The points on a straight line are representations of the reals. Do you > > > think so? > > > > No, I don't think they are "representations" of real numbers. In > > analytic geometry, in, for example, the plane R2, they are ordered > > pairs of real numbers. > > Now let one of the numbers of the pairs constant and forget it and we > meet. There are a lot more lines in R2 for which neither neither coordinate is constant, There are only two of infinitely many directions for which the lines are not skew. > > > > > > With Cantor you have to think that some points on the > > > straight line (to be exact: the absolute majority of them) are not > > > reachable by construction. > > > Should this be a consequence of an useful mathematics? > > > > Give me your axiomatization that is rich enough for ordinary calculus, > > and with a mathematical definition of 'constructible', and such that > > every real number is constructible. Then we'll talk about that. > > Do we really need such an axiomatisation? 99% of mathematics was done > without it. And was done right. But 99% is not good enough, in mathematics.
From: Virgil on 5 Oct 2006 03:59 In article <4c2fb$4524b769$82a1e228$28724(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > David R Tribble wrote: > > > Albrecht wrote: > > > >>>The idea of "all infinite many objects" in any sense is > >>>self-contradicting. There is no "all" of infinite many things. > > > > Han de Bruijn wrote: > > > >>Bravo! Another mathematician (?) who cares about disciplinary thinking. > > > > Let's see. If I define set N as containing "all" of the naturals, i.e., > > {0,1,2,3,...}, you're (both) saying that N can't exist as a real set, > > because there is no way to have "all" of an "infinitely many" elements > > at once. > > > > So then we must conclude that there are naturals that are not in N, > > because N can't contain all the (infinitely many) naturals. > > > > So now I define another set N' to contain all the naturals except those > > that are already in N, i.e., those naturals that we concluded must be > > omitted from N. > > > > So what is in this second set N'? Is it also not an infinite set? > > You have just demonstrated that infinitary set theory does not work. > > Han de Bruijn Wrong again! He has merely shown that HdB's interpretation of it does not work, but that only limits HdB's interpretation, not anyone else's.
From: mueckenh on 5 Oct 2006 07:07
Tonico schrieb: > Han de Bruijn wrote: > > Tonico wrote: > > > > > Han de Bruijn ha escrito: > > > > > >>Wishful thinking on the part of William Hughes. A simple Google search > > >>will reveal that the army of 'sci.math' dissidents is steadily growing. > > > > > > ************************************************** > > > Zaas! Dissidents...from sci.math???? Didn't know somebody already > > > formed a political or social or economical or > > > whatever-that-isn't-science group called sci.math, and that it already > > > has its dissidents! Perhaps Han means people that insist in talking > > > about mathematics with some mathematicians from a non-mathematical > > > point of view and without knowing mathematics? People that attack > > > mathematicians and even mathematics (go figure!) when someone dares to > > > point out some mathematical mistake in some nonsense that THEY say is > > > correct IN SPITE of evidence in contrary? > > > > One cannot speak of correct with a mathematics that is a non-discipline > > and gives non-discplinary answers to ill-posed questions like this one: > > > > http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#bv > > > > Zero balls at noon? Carl Friedrich Gauss would have turned in his grave. > > > > Han de Bruijn > > ************************************************** > For what we know, I think dead people don't turn in their grave, or > anywhere else for that matter. I agree though that the balls-vase-noon > is a question ill-posed but with possibilities to be pretty interesting > and deep into understanding some aspects of infinity and stuff. > I'd rather pose the next thinker. > Supose X is a person that never dies, and when he's 50,000 years old > he begins writing his autobiography That is not far from the story of Tristram Shandy. > in a rather peculiar way: every day > after his 50,000-th birthday he writes down one day of his life, > beginning with his first day of life. The question is: does X write > down ALL the days of his life in this autobiography? Pay attention: I > am not asking whether there will ever be a book containing all the days > of X's life, but rather whether there will exist some pages written by > X describing the events of some given day of his life, for EVERY > singled out day of X's life...? Why do you think this question be more important or interesting than the other? But the answer is easy: He never writes down the next day, where "next" is a number like "omega". > Of course, the above story is very similar to the one about the hotel > in space witrh infinite rooms in it, etc. > Regards > Tonio > Ps. BTW abiut the balls-vase matter, I think the best answer is the > one that said that noon, as we know it, is never reached according to > the spirit of the question. And the same answer is appropriate if considering the whole set N: It does not exist. Regards, WM |