An uncountable countable set
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From: mueckenh on 17 Jul 2006 09:40 Virgil schrieb: > > The successor of all naturals is not a natural and, therefore, must be > > larger (because it is not less). > > There is no such thing as the successor of all naturals any more that > there is a single successor common to both 3 a and 6. This is an expression coined by Cantor: "a number following after all natural numbers". I know tha it is nonsense. Therefore I quoted it. There is no number following all naturals and, hence, there is no number omega at all. > > One has the set of all naturals, and that set can have a successor under > the definition that the successor of any set, x, is (x union {x}). That is nonsense too. One cannot have a set of all naturals, because the step x U {x} defined by the axiom does never result in this set. You recently asserted that such stepwise processes can never exhaust a set. Now you changed your mind? Just in time? Regards, WM
From: mueckenh on 17 Jul 2006 10:00 David Hartley schrieb: > >> >> I cannot comment on the German text, but Dik's reading of the English is > >> >> clearly correct, WM's wrong. To put it even more clearly > >> >> > >> >> Question: Which transformations preserve ordinal number? > >> >> Question: What is an order-preserving transformation? > >> >> Answer: Those which are realised by a finite or infinite set of > >> >> transpositions. Answer: A finite or infinite set (at another place: sequence) of transpositions, each of two elements. > >> >> > >> >> "Transformation" is not defined here. > >> > > >> >But "such transformations which do not change the well-order" are > >> >defined here. And only those are interesting. > >> > >> They are not defined. They are characterised within the class of all > >> transformations, but transformations themselves are not described at > >> all. To use Cantor's proposition, to show that a particular > >> set-theoretic "object" is an order-preserving transformation, you would > >> need to show it can be written as a set of transpositions *and* that it > >> is a transformation. Nowhere - in the sections you've quoted - does he > >> say what a transformation is. He presumably knew what he meant; he may > >> well have written a definition somewhere else, but you do not seem to be > >> able to find it. > > > >Cantor was often redundant. Because he was very eager to get understood > >quite well. But he did never defined what "is" is or what "to write" > >means. Have you got me? In all of his collected works he does not > >define "Umformungen" because there is nothing do define. An "Umformung" > >is just a re-ordering. And such re-orderings which consist of a set of > >transpositions, were *defined* by him. > > As I and others keep reminding you, it doesn't make any difference what > Cantor actually meant. Quoting Cantor does not constitute proof. But outspoken wrong interpretations like that of Dik and yours must be corrected. >If you > want to make your supposed proof of inconsistency rigorous, you must: > > a) define what you mean by the result of applying an infinite sequence > of transpositions, Has been done. > > b) prove that your particular sequence re-orders a well-ordering of the > positive rationals to the usual ordering, > > c) prove that such a sequence cannot alter the order-type. It is impossible to have a dense order and a well order simultaneously. > > (Naturally, you will do this within a standard formulation of modern set > theory.) > > Dik and I think you can do b but not c. Virgil thinks you can't do b, > but until you provide a your claims are "not even wrong", just > meaningless. What would a proof mean? Dik would think this and Virgil would think that and you would perhaps have another opinion, but nobody would accept this as a proof, because all like set theory and then it would be dead. Set theory is meaningless, but it has the advantage that it cannot be wrong. Regards, WM
From: mueckenh on 17 Jul 2006 10:09 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > >> You may recall that every sequence member has trailing zeros. > > > > Indeed. I recall. Therefore the linearity of the list numbers enforces > > a column with zeros. > > Nice try, but non sequitur. There is no such column. Otherwise show one > or prove its existence. The proof requires logic. Therefore I am afraid you will not accept it. It reads: Either there is a column with only zeros, or there is at least one 1 in each column spanned by the digit positions of 0.111... , isn't it? But you will "argue": Nice try but there is neither nor. > > >> > As the first case, call it mixed number, is excluded by the > >> > definition of the list, > >> > there is only the > >> > second case remaining, call it linear number. > >> > >> This is also "excluded" by the definition of the list. So it's > >> pointless to discuss it either. > > > > Then the sum cannot be 0.111... Or "forall" must have a changed > > meaning (because, you know, there are no mixed numbers with zeros > > jumping around). > > But the *-sum _is_ as has already been formally shown 0.111.... > And no, forall does not have a "changed meaning". Consider the columns spanned by the digit positions of 0.111... Either there is a column with only zeros, or there is at least one 1 in each column, or ? > > Is it in particular point 4 which excites your fury? > > I will not read it. The ultimate defense. Regards, WM
From: Franziska Neugebauer on 17 Jul 2006 10:26 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> > >> >> You may recall that every sequence member has trailing zeros. >> > >> > Indeed. I recall. Therefore the linearity of the list numbers >> > enforces a column with zeros. >> >> Nice try, but non sequitur. There is no such column. Otherwise show >> one or prove its existence. > > The proof requires logic. Therefore I am afraid you will not accept > it. Your proof is free from logic? > It reads: Either there is a column with only zeros, or there is at > least one 1 in each column spanned by the digit positions of 0.111... > , isn't it? > But you will "argue": Nice try but there is neither nor. What you call a proof is an unproven proposition. >> >> > As the first case, call it mixed number, is excluded by the >> >> > definition of the list, >> >> > there is only the >> >> > second case remaining, call it linear number. >> >> >> >> This is also "excluded" by the definition of the list. So it's >> >> pointless to discuss it either. >> > >> > Then the sum cannot be 0.111... Or "forall" must have a changed >> > meaning (because, you know, there are no mixed numbers with zeros >> > jumping around). >> >> But the *-sum _is_ as has already been formally shown 0.111.... >> And no, forall does not have a "changed meaning". > > Consider the columns spanned by the digit positions of 0.111... > Either there is a column with only zeros, or there is at least one 1 > in each column, or ? Can't see that. How do you conclude this? What I see is: A column (coulmn e N & E row (row e N & a_row,column = 1)) (Every column contains a 1). >> > Is it in particular point 4 which excites your fury? >> >> I will not read it. > > The ultimate defense. Self protection. F. N. -- xyz
From: mueckenh on 17 Jul 2006 10:40
Dik T. Winter schrieb: > In article <nyAH0KfRjhuEFwtL(a)212648.invalid> David Hartley <me9(a)privacy.net> writes: > ... > > >> Works, p. 214, translated: > > >> The question, through which transformations the ordinal number of a > > >> well-ordered set is changed, and through which it is not changed, is > > >> easily answered, those, and only those transformations leave the > > >> ordinal number unchanged that can be rewritten as a finite or infinite > > >> set of transpositions, that is, of interchanges of two elements. > ... > > Question: Which transformations preserve ordinal number? > > > > Answer: Those which are realised by a finite or infinite set of > > transpositions. > > > > "Transformation" is not defined here. For that matter, neither is the > > action of an infinite set of transpositions, nor is there any proof. If > > you (WM) want to use this proposition in your argument, you must fill > > all these gaps. > > There is one thing more that bothers me about the quote. It speaks > about "can be rewritten as ... set of transpositions". What is the > precise meaning here? "Rewritten" is not the precise translation but "zurückführen" mans in this context simply: Such tranformations which consist of ... transpositions. > As transpositions in general do not commute, > I would not use the word "set" unless the set contains only > transpositions that mutually do commute. > > But as this is from letters by Cantor to Dedekind My first quote is not from the a letter to Dedekind but from a long paper "Über unendliche lineare Punktmannigfaltigkeiten". Cantor, Georg: "Über unendliche lineare Punktmannigfaltigkeiten", Math. Annalen 15 (1879) 1 - 7 Math. Annalen 17 (1880) 335 - 358 Math. Annalen 20 (1882) 113 - 121 Math. Annalen 21 (1883) 51 - 58 Math. Annalen 21 (1883) 545 - 591 Math. Annalen 23 (1884) 453 - 488 published as a book: Cantor, Georg: "Über unendliche, lineare Punktmannigfaltigkeiten", Teubner, Leipzig (1984) > (and not published > work, so probably research in progress) we likely never will know the > exact meaning. It is that simple that no explanation is required. Don't forget: Cantor speaks of a *simple* answer to the question. > But, whatever, the book with Cantor's letters to > Dedekind is in our library, so I will look sometime next week whether > more about that has been written. (I think it was Dedekind, but I > can check that too.) No it was Laßwitz. The definition of "such transformations" including "sequences" stems from a letter, 15. Febr. 1884, to Herrn Prof. Dr. Kurd Laßwitz in Gotha. It continues: Da nun bei einer endlichen Menge, wenn der Inbegriff ihrer Elemente derselbe bleibt, jede Umformung sich auf eine Folge von Transpositionen zurückführen läßt, so liegt hierin der Grund, warum bei endlichen Mengen Ordnungszahl und Kardinalzahl gewissermaßen zusammenfallen, indem hier Mengen derselben Valenz in jeder Form, als wohlgeordnete Mengen gedacht, immer eine und dieselbe Ordnungszahl haben. Regards, WM |