Cantor Confusion
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From: mueckenh on 31 Oct 2006 08:22 Dik T. Winter schrieb: > > Attention: 0.111... has only finite initial segments - and nothing > > more. Only those can be indexed. > > You keep stating that, without proof. That it has only finite initial > segments, I agree. Not the remainder. It is you who denies proof. Give an example of a digit which does not belong to a finite sequence. If you cannot do so, then every digit belongs to a finite sequence. The sequence 0.111... consists of every digit but not of more. Therefore, there is not need and no use of talking about infinite sequences. > > > > And I again note that the notation 0.111... > > > (in the decimals) has only meaning due to the definition of that notation. > > > > You need no transfinity to show that lim [n-->oo] 1/2^n = 0 and that > > lim [n-->oo] (1 - 1/2^n) / (1 - 1/2) = 2. Because that was known before > > transfinity was introduced. > > But I never argued that. But it was argued in connection with my infinite binary tree. > > You deny that omega is required to define 0.111.... But you insist that > > transfinite induction is required to write down the result of 1/9 in > > decimal representation? I did not yet know that this system is that > > difficult. > > I did not insist that something like that is required. I wrote: > > > > You need transfinity when you want to show that something that holds in > > > the finite case also is valid in the infinite case. Induction will not > > > show that 0.111... is rational, > > And indeed. Induction will *not* show that 0.111... is rational. Induction will show everything that *can* be shown. Up to every finite position 0.111...1 is rational. More is not possible. Irrational numbers don't exist, but that is another topic. Anyhow, in the binary tree there is no transfinity required. Regards, WM
From: mueckenh on 31 Oct 2006 08:25 Dik T. Winter schrieb: > In article <1162139168.281239.183260(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > You need not tell that to me. You should tell that to Wolfgang > > > Mueckenheim who insists that sqrt(2) does not exist because it is > > > impossible to know all the decimals in its decimal expansion. > > > > It does not exist as a number. It has no b-adic representation. It > > exists as a geometric entity. It is an idea. > > It exists as the continued fraction [1,2,2,2,2,...]. It also exists > as 1 in the base sqrt(2) notation. And it exists in many other disguising. But it does not exist as a number which can be put in trichotomy with all other numbers. Regards, WM
From: mueckenh on 31 Oct 2006 08:34 Dik T. Winter schrieb: > In article <1162157341.348431.26550(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1162068028.638690.242480(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > Approximately correct results. However, you need further analysis to > > > determine how approximate the result is. But that is another field of > > > mathematics (numerical mathematics) that derives the result based on > > > exact results, and either analysis the error in the calculated result, > > > or analysis the closeness of an initial problem that would have the > > > calculated result as exact result. The former is mostly done in > > > numerical analysis, the latter in numerical algebra. Read, for instance, > > > The Algebraic Eigenvalue Problem, by J. H. Wilkinson. But of course you > > > know that book because, almost certainly, you use methods for calculations > > > that are based on the methodology developed in that book. > > No comments on this, Wolfgang? No, Dik, I did not read hat book. I use only trivial classical mathematics like Algebra, Anaylsis, number theory. > > Do and enjoy your mathematics. I will not disturb you. I am not very > > familiar with those things, but they may be very valuable. > > In that case, why are you stating again and again that my mathematics is > wrong? The foundations are wrong. There is no uncountable set. For some arguments look here: http://www.fh-augsburg.de/~mueckenh/MR/Argumente.mht But all that does not prohibit to do mathematics as we are used to do. Proof: We all do it. And there are no problems. It is the same as with physics. Most engineers who construct cars and ships do not need relativity. Regards, WM
From: mueckenh on 31 Oct 2006 08:44 Dik T. Winter schrieb: > In article <1162157584.352015.276510(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > Euclides was apparently smarter than Cantor in this. He used the > > > parallel axiom (postulate) for something he could not prove from the > > > other axioms. > > > > But which is obviously possible and correct under special > > circumstances, while Zermelo's AC is obviously false under any > > circumstances. > > You just state so. This is nothing more than opinion. > > > > But when mathematicians are bothered about the parallel axiom have > > > problems with the justification of that axiom and reject it, you > > > aparently have no problems. But when mathematicians do the same with > > > the (implicit) well-ordering axiom from Cantor you appear to have > > > problems. Why? > > > > Because I can construct parallels (or see their absence) but I cannot > > see a well-order which is not definable. > > Indeed, what you do not see does not exist, and when others see it they > are talking nonsense. How more opinionated you can get? Everybody knows that a well order of the reals is not definable. Please let me know how the well order should be realized, because the prove of existence means that it can be realized. (The head line of Zermelos papers.) > > > > With the axiom of choice, it is possible to well-order the reals. > > > But it can also be shown that such a well-ordering can not be defined > > > by a formula in ZFC. > > > > How then can the well-ordering be accomplished? Zermelo proved tat it > > could be accomplished. > > Oh, well, just in another thread there is a discussion about the existence > proof and showing a method to calculate such a thing. An existence proof > does not necessarily imply a method to actually provide such a thing. > There is a mathematical proof that li(x) - pi(x) changes sign infinitely > often. In your finite world you will never see such a sign change. In my world it is not excluded, to my knowledge, to find as many sign changes as desired. (If Littlewood had also proven that the observation of any sign change was impossible, then I would not believe in his proof.) But in your world it is excluded to find a well order of the reals. Neither a defintion nor a list can help. That is a significant difference.
From: mueckenh on 31 Oct 2006 08:50
The Ghost In The Machine schrieb: > In sci.math, Dik T. Winter > <Dik.Winter(a)cwi.nl> > wrote > on Mon, 30 Oct 2006 02:21:06 GMT > <J7xFv6.ECF(a)cwi.nl>: > > In article <1162139168.281239.183260(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > You need not tell that to me. You should tell that to Wolfgang > > > > Mueckenheim who insists that sqrt(2) does not exist because it is > > > > impossible to know all the decimals in its decimal expansion. > > > > > > It does not exist as a number. It has no b-adic representation. It > > > exists as a geometric entity. It is an idea. > > > > It exists as the continued fraction [1,2,2,2,2,...]. It also exists > > as 1 in the base sqrt(2) notation. > > I am extremely puzzled as to why anyone would say that > sqrt(2) doesn't exist, given that numbers exist at all, > which is an abstract concept in its own right; however, > if 1 exists then N exists therefore J (or Z) exists > therefore Q exists therefore R exists (take one's pick: > Cauchy or Dedekind) therefore C exists. In my opinion a number should satisfy trichotomy with all other numbers. Let X = sqrt(2) and let Y be the number which you get by exchanging the digit at position 10^100 by 5. You will not be able to find out whether X < Y holds, because the univers has only less than 10^100 bits. And even if it was possible to determine that digit, you could never store the 10^100 first digits. sqrt(2) exists as the diagonal of the unit square and in many other shape, but not as a number. Regards, WM |