Cantor Confusion
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From: David Marcus on 29 Oct 2006 19:37 mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > And once again WM deliberately confuses, "all positions are > > > > finite", with "there are a finite number of positions". > > > > > > There is no confusing! Every finite position belongs to a finite > > > segment of positions (indexes). > > > > Correct. > > > > But > > every position belongs fo a finite set of positions, > > does not mean > > there are only a finite set of positions. > > > > The first means: > > > > for every position N, there exists a finite set > > of positions A(N) such that N belongs to A(N) > > > > The second means: > > > > there exists a finite set of positions B, such that > > for every position N, N belongs to B > > No it means only potential infinity: Every set B of positions is > finite, although there is no largest set B but every set B has a larger > superset B'. > Actual infinity would yield: There is a completed, finished infinite > set. That is nonsense, whther you call it finite set B or infinite set > |N. Finished infinity is a contradiction in itself, even worse than > quantifier changing. I don't think the word "nonsense" means what you think it means. > > You cannot simply exchange the quantifiers. > > It has nothing to do with that. If not, then please stop exchanging the quantifiers. > > Why. Why does "there are an infinite number of positions" imply > > that "at least one position is infinite"? > > Try to find an example for the opposite assertion: Try to construct an > actually infinite set with only finite numbers (which differ by a > constant value at least). Then you will know it. The set of natural numbers is an infinite set that contains only finite numbers. No idea what you mean by "construct" or "actually". Feel free to define them. -- David Marcus
From: Virgil on 29 Oct 2006 19:40 In article <1162157745.995131.292260(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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 French mathematical usage, particularly Bourbaki, uses "positive" and "strictly positive" where the correspponding English mathematical usage is, respectively, "non-negative" and "positive". Similarly for "negative' and "strictly negative" versus "non-positive" and "negative. When one is trying to move between the two languages, one must be very careful of which usage is the appropriate one if one is not to be mislead or to mislead others.
From: David Marcus on 29 Oct 2006 19:42 mueckenh(a)rz.fh-augsburg.de wrote: > > 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. Then why are you posting to sci.math? What do you possibly hope to achieve? > 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. -- David Marcus
From: David Marcus on 29 Oct 2006 19:45 mueckenh(a)rz.fh-augsburg.de wrote: > 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. What do you mean an axiom is "false"? Are you discussing philosophy or mathematics? -- David Marcus
From: Lester Zick on 29 Oct 2006 19:52
On 29 Oct 2006 13:35:46 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > >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. I have also heard that. In the early days of computing it was a significant issue as to whether zero was only positive or there was also a negative zero. These days zero is only regarded as positive and the so called negative zero is considered an overflow situation to the best of my knowledge. ~v~~ |