Cantor Confusion
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From: mueckenh on 29 Oct 2006 16:25 Virgil schrieb: > In article <1162069321.729271.190360(a)f16g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > Correct. And therefore no such thing can exist unless it exists in the > > > > > > mind. But we know that there is no well order of the reals. in any > > > > > > mind, because it is proven non-definable. > > > > > > > > > > It is perfectly definable, and perfecty defined, it is merely incapable > > > > > of being instanciated. > > > > > > > > Then let me know one of the perfect definitions, please. > > > > > > A set is well ordered when every nonempty subset has a first member. > > > > Which is the first member of the subset of positive reals? > > Under what ordering? if you claim an instanciation of such a well > ordering, present it. I do not. What kind of well ordering of the reals do you claim to exist? Defined? Catalogue? List? Else? (Please specify) Regards, WM
From: mueckenh on 29 Oct 2006 16:26 Virgil schrieb: > In article <1162069566.039416.155090(a)k70g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > But for infinite ones, you must do one > > > > > more transfinite step. > > > > > > > > No. > > > > > > Yes! > > > > In order to proceed from 1^/2^n to zero? > > In order to proceed in f(n+1) = 1*E + f(n)/2 to infinity? > > > > How could Cauchy calculate sqrt(2) by such a recursive procedure > > without knowing transfinite induction? > > By using ordinary induction, or by using continued fractions. Cauchy was > quite inventive and could no doubt have thought up other ways as well. Cauchy used f(n+1) = f(n)/2 + 1/f(n). When he came to sqrt(2) by induction, then I will come by induction to 2. > > > > > > Any real number has only finite digit positions, according to Dik > > > > > > And is Dik your "Authority"? > > > > No, but his answer shows that your party contradicts your own opinion. > > How do you deduce that because Dik and I agree that you are wrong that > Dik and I are necessarily agree on everything or are "of the same party"? > > This is not a two party system. No? According to my impression, it is. Regards, WM
From: mueckenh on 29 Oct 2006 16:29 Dik T. Winter schrieb: > In article <1162068028.638690.242480(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > William Hughes schrieb: > ... > > > Not according to you (at least not by the usual definition). > > > According to > > > your reasoning, there can only be a finite number of real numbers ever > > > defined. This makes limits kind of weird, since you cannot get > > > arbitrarily close. > > > > You cannot get arbitrarily close either. Can you calculate sqrt(2) to > > 10^100 digits? No. But we can both get as close as necessary to obtain > > correct results. > > 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. > > On the other hand, using approximations I can *never* determine that > sqrt(2) is a divisor of sqrt(13) + sqrt(37) in the algebraic integers. > Or have a look at <J7tKB0.MDo(a)cwi.nl>, using approximations you could > *never* derive such results. > > The whole point is to determine a number you do not need to determine > all the digits. You only give a method, or an indication, of > how the number is determined. Even this can onlky be done for a countable set. > Yes, I know that you do not want to > call those things numbers, but ideas. But mathematicians call them > numbers, and that is that. You have to live with the terminology. > And you have never given a proper definition of what *you* regard as > number. So until such a definition is forthcoming, I would think that > it is better to use the mathematical terminology. > > And when you think there are no applications for those numbers (ideas), > I can put your qualms to a rest. One of the basic problems in mathematics > is factorisation of integral numbers; much of cryptology depends on the > difficulty. One of the most succesfull methods (NSF, the Number Field > Sieve) and its derivatives depend just on such numbers (ideas). Do and enjoy your mathematics. I will not disturb you. I am not very familiar with those things, but they may be very valuable. I would not care at all unless these "numbers" were used in Cantor-lists as if all digits could be determined. Regards, WM
From: mueckenh on 29 Oct 2006 16:33 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. > > 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. > > 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. Regards, WM
From: mueckenh on 29 Oct 2006 16:35
Dik T. Winter schrieb: > "positive numbers" with "numbers larger than 0". Because I understand the > main domain is Anglo-Saxon mathematics. This is in contradiction to what > I did learn at university (0 is both positive and negative). Really? I never heard of that. Is here anybody who learned that too? It would interest me. No polemic intended. Regards, WM |