An uncountable countable set
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From: mueckenh on 9 Jul 2006 17:15 Virgil schrieb: > Cantor's proof avoids this need to "reach the end" by giving a general > rule which works independently for each n in N without reference to any > other member of N. This allow the simultaneous creation of all the > digits. Cantor's proof allows to find the digit number n for every n. My proof allows to find the transposition number n for every n. Cantor's well-ordering of the rationals has to delete many rationals like 2/2 and 6/3 or 3/6. In order to define the position of a certain rational in the well-ordering, the deletion must be done by hand. This problem becomes even more pressing when well-ordering the algebraic numbers. All this stuff must be executed successively. And now you find that difficult? Now you want to make us believe that in my well-determined sequence of transpositions the ordering might be mixed up? Poor arguing! > Because of the non-commutativity of permutations, your construction > cannot use this shortcut. In order to give a universal law, I do not need a shortcut. But I gave one recently: (1,2) (2,3) (1,2) (3,4) (2,3) (1,2) .... There is no limit process. The *only* criterion about manipulations on countable infinite sets is whether one can determine *precisely* at which natural number something happens: After how many steps in a well-order of |Q the fraction 4711/235537 will appear, for instance, or in which line of Cantor's list a certain diagonal element will be placed and so on. And I can determine *precisely* after how many steps the number 4711/235537 will be inserted in the order by magnitude with all of its predecessors of the initial well-order. This is fixed and can be calculated for any rational number. Therefore all rational numbers are covered and will successively appear in the well-order. The argument that there remain always infinitely many other rationals is wrong, because by definiton the fate of each and every rational is determined and can be calculated. It is not necessary to really carry out any transposition. Regards, WM
From: mueckenh on 9 Jul 2006 17:19 Virgil schrieb: > Not so. Cantor's rule works for each n independently of all other > members of N, so all can be done simultaneously by a single rule. > > Yours does not allow simultaneity because of the non commutativity of > permutations. Nonsense! I do not need and do not want to commute any transpositions. And the result is determined for any rational of the initial well-order. The argument that there remain always infinitely many other rationals is wrong, because by definiton the fate of each and every rational is determined and can be calculated. It is not necessary to really carry out any transposition. > If you do it Cantor's way, all steps are done at once. No step is done. The well -order is determined by the rule at once. That is all. Regards, WM
From: mueckenh on 9 Jul 2006 17:25 Dik T. Winter schrieb: > > 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. And that is all you have to object against my proof? We have a fixed scheme of transpositions. But we could apply the transpositions even without any law, applying only the rule that a pair which is already ordered shall not be treated for a second time. Then we can apply countably many arbitrary transpositions until all elements are ordered by magnitude. The limit to which every path will lead is always the same: The set of rationals ordered by magnitude and by natural indices. But there is no limit process. The *only* criterion about manipulations on countable infinite sets is whether one can determine *precisely* at which natural number something happens: After how many steps in a well-order of |Q the fraction 4711/235537 will appear, for instance, or in which line of Cantor's list a certain diagonal element will be placed and so on. And I can determine *precisely* after how many steps the number 4711/235537 will be inserted in the order by magnitude with all of its predecessors of the initial well-order. This is fixed and can be calculated for any rational number. Therefore all rational numbers are covered and will successively appear in the well-order. The argument that there remain always infinitely many other rationals is wrong, because by definiton the fate of each and every rational is determined and can be calculated. It is not necessary to really carry out any transposition. > > > > 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. It is needed to show that the diagonal > number is a real number. But that is not interesting. It is easy to see that the diagonal is a sequence of the same sort as are the list entries. Whether they are real numbers is uninteresting. Interesting is that such sequences are uncountable. But in order to prove that the diagonal differs from every entry, there a limit is required but not available. >>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). The missing limit has not been remedied but has only been put aside. My objection remains: 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. > 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. Sind nämlich m und w irgend zwei einander ausschlie�ende Charaktere, so betrachten wir den Inbegriff M von Elementen E = (x1, x2, ..., x�, ....), welche von unendlich vielen Koordinaten x1, x2, ... x�, ... abhängen, wo jede dieser Koordinaten entweder m oder w ist. I think one can safely interpret this sentence as describing binary numbers, though they were not called so. Regards, WM
From: mueckenh on 9 Jul 2006 17:33 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > The set ordered by size and by natural indices is the limit of my > > transpositions. > > 1. What is a limit of transpositions? > 2. How do you prove its existence and uniqueness? How does Cantor prove the existence and uniqueness of the well-ordered set of rationals or of the diagonal number of his list? The *only* criterion about countable infinite sets is whether one can determine *precisely* at which natural number something happens: After how many steps in a well-order of |Q the fraction 4711/235537 will appear, for instance, or in which line of Cantor's list a certain diagonal element will be placed and so on. And I can determine *precisely* after how many steps the number 4711/235537 will be inserted in the order by magnitude with all of its predecessors of the initial well-order. This is fixed and can be calculated for any rational number. Therefore all rational numbers are covered and will successively appear in the order by magnitude while remaining in the well-order. By definiton the fate of each and every rational is determined and can be calculated. It is not intended to really carry out any transposition. Regards, WM
From: mueckenh on 9 Jul 2006 17:42
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > >> 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. Excuse me. There is not a single exchange carried out. All aere simultaneously defined. More is not required. Have you ever tried to well-order the set of rationals for more than, say, 20 elements. Nobody does carry out such exercise. The well order is given by the principle. The order by magnitude is given by my principle too. > > 1. There is no "exchance". > 2. d_i def= f (a_ii) is done A i e N. There are no "list numbers left". You are right. I responded only to Virgil's very naive belief that there something really should be carried out. Of course it is sufficient to give the principle by a short expression like this (1,2) (2,3) (1,2) .... to intelligent people, and all is instantaneously ordered by magnitude. Regards, WM |