An uncountable countable set
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From: Virgil on 8 Jul 2006 16:46 In article <1152384783.335604.190890(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Hartley schrieb: > > > In message <vmhjr2-E08407.11521207072006(a)news.usenetmonster.com>, Virgil > > <vmhjr2(a)comcast.net> writes > > >At best you have proved that a countable subset of the rationals have > > >ben restored to their usual ordering, but you have nowhere shown that > > >your process ever re-orders ALL of them. > > Cantor has nowhere shown that his definition covers all natural > numbers. He has to me, and to all those who know how to read his definition. If he has not to you, the fault lies in you and not in Cantor. > > > > > >After each of your countably many transpositions, there is still a > > >countably infinite subset of the rationals left which are not in natural > > >order. That is a necesssary consequence of the entire set being > > >originally well ordered. > > But in the moment of the definiton, the result is determined for all > rationals as the well-order is determined for all rationals, though in > fact only a small set can be written. It is the principle which counts. > Therefore your arguing is invalid. In matters of logic, "mueckenh" does not seem to have any principles. What "principle" does "mueckenh" allege he is counting on to establish his counterfactual claims? > > And it will be covering all rationals, as soon as the transposition > principle is given. > > > > (WM's fallacy is his claim that the (order-preserving) indexation by the > > naturals carries over to the limit.) > > Cantor's fallacy is that the indexation by the naturals carries over to > the limit. Cantor does not need any limit process, his rule works simultaneously on all n in N. > > Regards, WM
From: Dik T. Winter on 8 Jul 2006 22:48 In article <1152384019.621477.88880(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > > And what is the consequence of this? > > > > > > > > That the calculation of the digits can be done in parallel? > > > > > > It is done in zero time. It is determined from the beginning. Why > > > should something "be done"? > > > > It was you who remarked on the calculations... > > The result of a calculation does not depend on the time you need to > obtain it. No, but it may depend on the ordering of the calculations. So there is an inherent difference between calculations that are independent of each other and calculations where some calculation can not be done before all previous calculations are done. So when calculating the digits of the diagonal number it does not matter what digit you calculate first. In your re-ordering sequence it makes a strong difference what re-ordering you do first. > > > But he needs to consider every digit with equal weight. That is not a > > > limit process. > > > > What did he *mean* when he wrote "with equal weight"? > > He did not write that. I used this description to express that all > digits must be distinguishable from the exchanged digits and that the > resulting number must be distinguishable from the number with one digit > left unexchanged, be it the first or the last one (which can be > recognized). Yes, that is what I wrote in some of my articles (that you have read). The limit process is *not* needed to distinguish the diagonal number from all other numbers of the list. I is needed to show that the diagonal number is a real number. > By the way, Virgil's arguing leads to the "paradox": There is no last > number and not that one next to the last and so on. Which one is the > first one that does not exist? Eh? Care to explain? What is the paradox? There is no last numbers. Hence, there is also no predecessor of the last number. That seems pretty clear to me. There is no predecessor of something that does not exist. What makes you think that there should be a first one that does not exist? This is (to me) close to gibberish. > > No. You do *not* and he does *not*. In the Cantor diagonal the number > > obtained is a real number using limits (in the Cauchy sense) with the > > definition of real number by many others (that can be proven to all be > > equivalent with each other). The limit used by Cantor is precisely the > > epsilon argument, together with majoration and minoration on the ordered > > set of rational numbers. By Cauchy any sequence of decimal digits has a > > limit, but it is not certain whether that limit is in the defined set of > > numbers. By Cantor, Dedekind, Weierstrass and a host of others (using > > various formulations), such a limit (when starting with rational numbers) > > is defined as a real number. That is the place where Cantor uses the > > limit (although he may not have expressed it as such), i.e. showing that > > the resulting number is a real. > > > Cantor does not at all use real numbers but merely infinite sequences > with only two different symbols w and m (which might be interpreted as > binary representations but were not). But that there is no satisfactory > limit consideration becomes clear from the following: We know that > 0.999... = 1.000... This leads to the result that a change of 1 in the > limit where the digit number goes to oo does not have the effect which > would be required in order to distinguish the diagonal number from the > list numbers. You still keep with Cantor's original papers. Perhaps you are right. But since that time the arguments have been improved (and more understood). Arguing against Cantor's original papers is futile. But let me take one point. You write: > with only two different symbols w and m (which might be interpreted as > binary representations but were not). And you continue with taking them to be binary representions. This is dishonest. You state, explicitly, that they were not binary representations, and argue against them as if they were binary representations. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 9 Jul 2006 08:50 Dik T. Winter schrieb: > In article <1152282125.757193.28320(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > Therefore: As my sequence of transpositions > > (1, 2) > > (2, 3) > > (1, 2) > > (3, 4) > > (2, 3) > > (1, 2) > > ... > > determines exactly which transposition will be executed as the n-th, > > everything is determined and everything is finite. > > > > > > And again, it maintains the well order as long as you remain in the finite > > > domain. So up to every finite n, > > > > THERE IS NO OTHER n ! ! ! > > Indeed. > > > > you have ordered the first n elements > > > of the well-ordered list, retaining a finite list. And back again to > > > problem 1 with this. How do you define that when n grows without bound? > > > > Also without bound n is always finite, or it would not be a natural > > number! > > Ok, good, so your sequence of transpositions will never give the standard > order of the rationals (as you claimed). This is correct, but it is correct too in case of Cantor's list, even if it is completed in zero time. > > > > How do you define the "limit"? And if you define that, is that "limit" > > > also well-ordered? Those are things you have to prove. > > > > There is no limit other than in Canrtor's diagonal. EVERY set initially > > indexed by natural numbers will remain indexed by natural numbers. And > > it will unavoidably become ordered by size too. > > There *is* a limit, as I wrote already: The set ordered by size and by natural indices is the limit of my transpositions. > > > > > But Cantor's arguing is that without epsilon. > > > > > > I do not know Cantor's argument exactly. I think that he implicitly > > > uses that result. In my formulation it was abundantly clear that I did > > > use it. > > > > Cantor argues: If for all natural numbers something is defined (like > > the bijection n <--> 2n or the replacement of 5 by 4) then there is no > > further reason for any limit process. Cantor in his paper about the > > diagonal argument does not at all consider any limit process. > > How then does he show that the diagonal he obtains is a real number, without > (implicitly) using a limit? He does not at all talk about real numbers, but about a Mannigfaltigkeit of w's and m's and a sequence which he, as customary at his times, calls a series (Reihe). Contrary, he avoids numbers: "Es läßt sich aber von jenem Satze ein viel einfacherer Beweis liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist." There is no limit consideration and there cannot be a limit consideration because *every* Cauchy-epsilon must leave infinitely many digits undetermined. Otherwise you would have to find an eypsilon which covered only the last digit. And even then this very last digit was uncertain. Therefore the epsilon must even by smaller than 10^(-largest n), which is impossble as you know, because the largest n is lacking. The idea that Cantors diagonal was defined as a nunber accrding to Cauchy AND that each digit was well distinguished from the digit of the corresponding list-number is a wish, an imagination, a fairy tale. Regards, WM
From: mueckenh on 9 Jul 2006 08:55 Dik T. Winter schrieb: > Right. But this does *not* prove that there is an n that is equal to K. > To reformulate clearly > For all p there is an n such that An[p] = K[p] > this does *not* imply that > There is an n such that for all p An[p] = K[p], > which is what you are arguing. > Take as an example of a finite list n 0.010 0.011 0.101 and K = 0.111 with p = 1,2,3 Ap En : n = p does not imply K is in the list But if you have a linear increasing sequence n 0.100 0.110 0.111 and K = 0.111 with p = 1,2,3 Ap En : n = p implies K is in the list My proof covers the latter case in the limit of all natural numbers. By my construction we have the fact that there is an An in the list A1 0.1 A2 0.11 A3 0.111 .... such that for all p An[p] = K[p], if p is a natural number. Of course we cannot determine this An. But it is in the list, because this list is the complete list of all natural numbers. > Pray explain how you come (in logical steps) from the first statement > to the second. Your logical steps are only valid for the case that the An in question could be named. That is of course impossible. But in a complete list of naturals it is present by definition. Regards, WM
From: mueckenh on 9 Jul 2006 08:56
Virgil schrieb: > In article <1152282125.757193.28320(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > There is no limit other than in Canrtor's diagonal. EVERY set initially > > indexed by natural numbers will remain indexed by natural numbers. And > > it will unavoidably become ordered by size too. > > At best you have proved that a countable subset of the rationals have > ben restored to their usual ordering, but you have nowhere shown that > your process ever re-orders ALL of them. Cantor has nowhere shown that his diagonal ever covers the whole list. > > After each of your countably many transpositions, there is still a > countably infinite subset of the rationals left which are not in natural > order. That is a necesssary consequence of the entire set being > originally well ordered. After each of his countably many exchanges there is still a countably infinite subset of the list numbers left which are not yet exchanged. Regards, WM |