An uncountable countable set
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From: Dik T. Winter on 9 Jul 2006 20:23 In article <1152450704.938408.69690(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Virgil schrieb: > > What we have said, and what is quite true, is that for every natural > > there is a larger natural. > > Of course. And therefore it is impossible to exhaust all of them or to > find a set which is larger than all naturals together. But nowhere an attempt is made to exhaust them. There is only an axiom from which it can be derived that there is a set that contains them all. With that axiom, Cantor's argument is a proof. Without that axiom, Cantor's argument is meaningless. You are not arguing against Cantor's argument, you are arguing against that axiom. > > Right! There are more than any finite number of finite naturals, indeed, > > an endless supply of them which collectively form an infinite set. > > There is no "supply". The elements of a set do exist, instantaneously > and immediately. Sets are static. Yup. So an axiom was needed to assert the existence of such a set. -- 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 9 Jul 2006 21:54 In article <1152480309.668400.137270(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > 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. It is my objection to what you see as the similarity. > 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. I thought that was part of the rule (they are really conditional transpositions), so what? > 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. Again, you use the word "limit" here, without definition. How do you *define* the limit of a sequence of transpositions? How do you *define* the limit of the set on which the transpositions are applied? > 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, There are no "steps" in a well-order of Q. There is no notion of at which natural number something happens. A well-ordering of Q means a precise set of rules that determine the natural number to which a rational corresponds. It is your re-ordering that requires some notion of limit. > for instance, or > in which line of Cantor's list a certain diagonal element will be > placed and so on. This makes no sense to me. In the first place, it is not Cantor's list, it is a given list. In the second place I have no idea what you mean with "diagonal element". But what is known is that the n-th digital place of the diagonal is derived from the n-th element in the given list. > 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. Indeed, you can. But now you are talking about steps. In the previous no steps were involved. So, you are talking about a sequential process, which was not he case in the previous things. A mapping that well-orders the rationals is *not* a sequential process. The determination of the n-th digit of the diagonal from a given list is *not* a sequential process. > 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. Yes, for each rational number in the well-ordered list you can calculate the step when it comes in place in a numerically ordered segment of the rationals. But this does *not* mean that the final result is a well-order. Because at no time can you calculate the place where that rational number will be at the end. You need to show (at least) that there is a first element in the final ordering. > > 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. But it is just that part that is interesting. Try the same with a sequence of algebraic numbers. You need to prove that what you get is also an algebraic number (and you cannot). So for algebraic numbers the proof fails. For real numbers the proof goes through, because you can prove that the resulting number is also a real number. > Interesting is that such sequences are > uncountable. A sequence is *never* uncountable. By definition of the words sequence and uncountable in mathematics. > But in order to prove that the diagonal differs from every entry, there > a limit is required but not available. No, it is just that place where a limit is not required. By definition, for every n in N, the n-th digit in the diagonal will be different from the n-th digit of the n-th element of the list. I do not see why limits are needed here. > 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... Yes, that is because of the way those notations are *defined*. Those notations do not come out of thin air. They need definition before they can be used. And their definitions include limits. > 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. I can make no sense of this. > > > 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
From: Virgil on 9 Jul 2006 23:06 In article <1152481681.567374.197070(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > >> What we have said, and what is quite true, is that for every natural > > >> there is a larger natural. > > > > > > Of course. And therefore it is impossible to exhaust all of them or to > > > find a set which is larger than all naturals together. > > > > What exactly does "larger than all naturals together" mean? > > Uncountable. Nonsense. What "mueckenh" is trying to say is that the axiom of infinity should not work. But it does in spite of him. > > > > >> Right! There are more than any finite number of finite naturals, > > >> indeed, an endless supply of them which collectively form an infinite > > >> set. > > > > > > There is no "supply". The elements of a set do exist, instantaneously > > > and immediately. Sets are static. > > > > Sudden insight? > > Clever application of different standpoints. If there is no supply of naturals, then that are no naturals. But if there is a supply of any naturals, ther is a set of all of the. So the set of all naturals is static. So what? No one expects it to be anything else, as far as I know. > > Regards, WM
From: Virgil on 9 Jul 2006 23:08 In article <1152481853.625560.325970(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > All positions are indexed <==> K is in the list. It is a logical > equivalence for linear sets like my list. It that "mueckenh"'s fairy tale set K? If not what is it?
From: mueckenh on 10 Jul 2006 05:21
Virgil schrieb: > In Cantor's 'diagonal" proof, no numbers are transposed. > It is merely shown that a rule of construction can be stated such that > for an arbitrary list of reals and an arbitrary member of that list the > constructed number is not equal to the selected number. This is only shown for a finite set of n numbers. It can only be proved after having counted the digits from 1 to n without leaving out a single one. It leaves always an infinite (and in reality *uncountable*, though called "countable") set of numbers. > From which one may deduce that a complete listing of reals is not > possible. If one decides to belong to the set of those blind people which call the uncountable "countable", which call the not existing "existing", and which call themselves logicians without having a clue what logic is (compare the binary tree). Regards, WM |