Cantor Confusion
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From: Lester Zick on 29 Oct 2006 14:48 On Sun, 29 Oct 2006 00:32:15 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >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. I'm inclined to disagree. > 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. Who says? >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. A number is a straight line segment. Rational and irrational numbers are defined this way. Transcendental numbers are defined on curves which have no exact correspondence to straight line segments. >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). ~v~~
From: Virgil on 29 Oct 2006 13:49 In article <1162140211.609455.109800(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > 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). > > Most know a bit set theory, but in their study they do not take curses > in set theory. In Germany it is not obligatory. I doubt that is > different in other countries. 90 % will not be able to tell the ZFC > axioms when asked. Most of them will not know that there is no > definable well order of the real numbers. Many will not even know what > a well order is. They all can be very good mathematicians, and in > general they are. If they are all that good, then they will quickly see the self-contradictions in WM's theses when presented with them.
From: Lester Zick on 29 Oct 2006 14:51 On Sun, 29 Oct 2006 13:03:00 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> >> David Marcus 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). If you don't believe that and assert >> > > the contrary, then try to find a finite position which dos not belong >> > > to a finite segment (of indexes). >> > > >> > > It is simply purest nonsense, to believe that "all positions are >> > > finite" if "there are an infinite number of positions". >> > >> > Before any of us wastes more time on this, please pick one: >>> >> > 1. You are making a statement that you say is provable within some >> > standard mathematical system, e.g., ZFC. >> > >> > 2. You are making a statement that is true within your own system. >> >> My above statement is true within any system which is free of self >> contradictions. > >That's nice, but it isn't what I asked. I'll try again with a more >specific question: > >1. Is your statement provable within ZFC, i.e., using the axioms and >rules of inference of ZFC? Is ZFC free of self contradictions? ~v~~
From: Lester Zick on 29 Oct 2006 14:53 On Sat, 28 Oct 2006 17:52:32 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >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). If you don't believe that and assert >> the contrary, then try to find a finite position which dos not belong >> to a finite segment (of indexes). >> >> It is simply purest nonsense, to believe that "all positions are >> finite" if "there are an infinite number of positions". > >Before any of us wastes more time on this, please pick one: Anything you contribute to the subject could hardly be a waste of your time. >1. You are making a statement that you say is provable within some >standard mathematical system, e.g., ZFC. > >2. You are making a statement that is true within your own system. ~v~~
From: David Marcus on 29 Oct 2006 14:58
Virgil wrote: > In article <1162139168.281239.183260(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > 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. > > Being a "number" is nowhere defined as having to have a complete decimal > expansion, except possibly by WM himself, but everyone knows how screwed > up he is. > > Real numbers have two standard definitions generally accepted: > (1) as Dedekind "cuts" > (2) as equivalence classes of Cauchy sequences of rationalsmodulo the > null sequences. You can also construct the real numbers as infinite decimals. > WM has no generally accepted definition, so they are particular to him > alone, and are of no weight elsewhere. -- David Marcus |