Cantor Confusion
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From: David Marcus on 29 Oct 2006 20:56 Lester Zick wrote: > 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. You've jumped from Mathematics to Computer Science. IEEE floating point does provide minus zero. It is useful for dealing with underflow (not overflow). -- David Marcus
From: Dik T. Winter on 29 Oct 2006 21:17 In article <1162137834.999423.231310(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > No. Any real number has only finite digit positions, according to Dik, > > > who rigorously denies that 0.111... cannot be indexed completely by > > > natural indexes. Now you see that this opinion leads to induction and > > > to a contradiction. So you hurry to switch to transfinity. > > > > Pray. No, you are wrong. 0.1 is rational, so is 0.11, and so on, so > > for every *finite* n, 0.1...1 (n-times) is rational. This does *not* > > show that 0.111... is rational. > > All 1's can be indexed in 0.1 and in 0.11, and so on. This does *not* > show that all ones in 0.111... can be indexed. Back again to that? For all n in N, the n-th digit is 1. > > 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, it can only show that all the finite > > initial parts are rational. > > In Germany children learn how to divide 1 by 9. After having gotten > that, they know hat 0.111... is rational. So, please show that 0.101001000100001... is rational. (The n-th digit is 1 if n equals k.(k+1)/2 for some k, it is 0 anywhere else.) > 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. > > 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 let's go on to continued fractions: [1,2,2,2,...] each (finite) approximant is rational. Nevertheless, the limit (yes, it can be properly defined) is not rational (it is sqrt(2)). So any arguiing about finite initial sequences can be wrong when we go to the infinite. You need no transfinity to show that the above limits are what they are, because the limit concept does not use 'infinity'. So there is no infinite case. > 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. As it also will not show that [1,2,2,2,...] is rational. But have a look at O. Perron, die Lehre von den Kettenbr?che, Berlin, 1913. a major work on continued fractions. -- 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 29 Oct 2006 21:21 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. -- 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 29 Oct 2006 21:23 In article <1162140211.609455.109800(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > 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. In the Netherlands, set theory was a required course for all undergraduate mathematicians. -- 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 29 Oct 2006 21:39
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? > > 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. But Wilkinsons analysis can only be done on an uncountable set. > > 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. In that case, why are you stating again and again that my mathematics is wrong? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |