An uncountable countable set
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From: mueckenh on 26 Jun 2006 06:56 Virgil schrieb:> > > Virgil. Such proofs must be wrong in general, and even the allmighty > > could change this fact. Should read: could not change this fact. Regards, WM
From: mueckenh on 26 Jun 2006 07:01 Daryl McCullough schrieb: > >Daryl McCullough schrieb: > > > >> mueckenh(a)rz.fh-augsburg.de says... > >> > >> >Such a function exists from |N --> |R for all real numbers which can be > >> >individualized, i.e. which are really real numbers. > >> > >> If such a function exists, then demonstrate it. Give us a function > >> f from N to R such that every real number that is "individualized" > >> is in the image of f. > > > >Cantor's general proof breaks down > > I'm not asking about Cantor's proof. I'm asking you to demonstrate > a function f from N to R such that every "individualized" real number > is in the image of f. Can you demonstrate such a function? If not, > why not? The diagonal number is not an individualized number, unless it is of such a simple kind I demonstrated. A random Cantor list can never be read to the bottom, the diagonal number can never be individualized. Give me a list of all numbers which you can really individualize. It is a complete list. Regards, WM
From: mueckenh on 26 Jun 2006 07:05 Virgil schrieb: > > > Which of the real numbers so determined are not really real numbers? > > > > Irrationals, for instance, are not numbers at all, because Cauchy's > > epsilon cannot be made arbitrarily small. > The members of the sequence, 1, 1/2, 1/4, 1/8, ... are all positive > but contain members smaller than any given positive , thus rationals can > be made arbitrarily small. And natural can be made arbitrarily large. But not all of them do exist. Take [pi*10^10^100], change the last digit to 6. You will not be able to find out whether one of both natural numbers is larger than the other. They do not exist. Nevertheless 10^10^10000 is an existing natural number. > > > But we will keep this > > disussion at the reordering of the well-ordered set of rationals. If > > you have understood that there are not all rationals, you will > > understand easily, that there is no irrational. > > In my imagination, all of the rationals exist and all of the irrationals > as well. I am sorry to hear that "mueckenh"'s imagination is so > self-limiting. It recognizes the reality. Regards, WM
From: Daryl McCullough on 26 Jun 2006 09:10 mueckenh(a)rz.fh-augsburg.de says... >Daryl McCullough schrieb: >> I'm not asking about Cantor's proof. I'm asking you to demonstrate >> a function f from N to R such that every "individualized" real number >> is in the image of f. Can you demonstrate such a function? If not, >> why not? > >The diagonal number is not an individualized number, unless it is of >such a simple kind I demonstrated. A random Cantor list can never be >read to the bottom, the diagonal number can never be individualized. Forget about Cantor for now. I'm asking you to demonstrate a function f from N to R such that every "individualized" real number is in the image of f. Can you demonstrate such a function? -- Daryl McCullough Ithaca, NY
From: mueckenh on 26 Jun 2006 10:01
Dik T. Winter schrieb: > > p=2E 214: "Die Frage, durch welche Umformungen einer wohlgeordneten Menge > > ihre Anzahl ge=E4ndert wird, durch welche nicht, l=E4=DFt sich einfach so > > beantworten, da=DF diejenigen und nur diejenigen Umformungen die Anzahl > > unge=E4ndert lassen, welche sich zur=FCckf=FChren lassen auf eine endliche > > oder unendliche Menge von Transpositionen, d. h. von Vertauschungen je > > zweier Elemente." > > Cantor admitted infinitely many transpositions. I think he meant > > countably many. But more aren't needed. > > No he did not. Pray read again. He was talking about the kind of > transformations of well-ordered sets for which the *number* of elements > would change. Or is well-orderedness part of the number of elements? Number (Anzahl) is only defined for well-ordered sets. Number (Anzahl) here is *not* meant as cardinality. The previous page reads: (Collected works, p. 213): "Durch Umformung einer wohlgeordneten Menge wird ... _nicht_ ihre _Mächtigkeit_ geändert, wohl aber kann dadurch ihre _Anzahl_ eine andere werden. Compare the letter from Cantor to Mittag-Leffler (17.12.1882): "Den größten Vortheil gewinne ich jetzt durch die Einführung ... der "Anzahl einer wohlgeordneten Menge".... Die Anzahl einer wohlgeordneten Menge ist also ein Begriff der in Beziehung steht zu ihrer Anordnung; bei endlichen Mengen findet er sich offenbar als unabhängig von der Anordnung; dagegen jede unendliche Menge verschiedene Anzahlen im Allgemeinen hat, wenn man sie auf verschiedene Weise als "wohlgeordnete" Menge denkt. Hence "Anzahl" is connected with the special kind of well-ordering. In particular different wel-orderings lead to different numbers (Anzahlen). > I think not. And indeed any number of interchanges can be performed on > a set without changing the number of elements. > > I think what he is arguing here (about a field that needed new exploration) > that at least for well-ordered sets any number of transpositions (and in > fact any finite or infinite permutation) would not change the number of > elements. Apparently it was not clear to him if that was also the case > for sets that could not be well-ordered. That is the best I can make out > of your quote. He may have thought as follows: One transposition does not destroy the well-ordering. n transpositions do not destroy the well-ordering. Infinitely many will not destroy the well-ordering, because there is no first one, which did so (as Virgil argued correctly). But, independent of what were Cantor's actual thoughts, if infinity would exist, we could order the rationals by magnitude and maintain the well-order. Regards, WM |