Cantor Confusion
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From: Virgil on 28 Oct 2006 19:01 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.
From: Virgil on 28 Oct 2006 19:07 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. > > > > 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.
From: David Marcus on 28 Oct 2006 19:37 Virgil wrote: > In article <1162067841.239007.74250(a)e64g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Please name the mathematicians that agree that your argument is correct. > > > (Han doesn't count, since he says he is a physicist.) > > > > Every mathematician whose mind has not yet suffered the drill of set > > theory you probably were exposed to knows that my argument is correct. > > "Mueckenheim" cannot speak for all those people. The set of mathematicians who don't know any set theory is probably pretty small. So, Mueckenheim's statement might be true (vacuously). > There are, no doubt, some of them that will come to understand how wrong > "Mueckenheim" is even without the benefits of formal education. -- David Marcus
From: Dik T. Winter on 28 Oct 2006 20:32 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 how the number is determined. 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). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 28 Oct 2006 20:52
In article <1162069321.729271.190360(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Virgil schrieb: .... > > > > > 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? With the common order the reals are not well-ordered, so there is none, and there is no contradiction. > If you can't > answer: How would you order a set? If you can't define an order, what > is your alternative? Writing a countable list? Scarcely sufficient for > R. An uncountable catalogue? Difficult to print. The question is, can the reals be well-ordered? Cantor assumed that all sets could be well-ordered. But it has later been shown that that could not be proven from the axioms he used (yes, I use that term again, because I think it is the most proper translation). An additional axiom was needed to have that the reals can be well-ordered. 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 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? 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |