An uncountable countable set
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From: Dik T. Winter on 16 Jul 2006 19:25 In article <1153051856.174981.269510(a)35g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > David Hartley schrieb: .... > By the way, why should Cantor have written that text, if it did not > express something? Ask Cantor. > > 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. > > That kind of arguing shows why set theory cannot be contradicted. It shows that when you can not provide a proof that there is a contradiction there is no clear contradiction. To proof a contradiction in a system with a certain set of axioms you have to show that it is possible to prove two contradicting things in that theory. If there is proof of one thing but no proof of another thing, a contradiction is clearly not shown. -- 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 16 Jul 2006 19:35 In article <1153052741.864854.24540(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > there is one natural number missing: 0.0... > > > > > > That is not a natural number. > > > > Depends on whether you are a follower of Bourbaki and Halmos, or of others. > > My learning was with Bourbaki as a basis. > > 0 was invented less than 2000 years ago. Natural numbers need not be > invented. However, Bourbaki and Halmos tried this trick in order to > prevent set theory from too easily been demasked as inconsistent. Eh? I was not talking about invention but about definition. What problems do you have with definitions? > > > The Addition > > > *+ was defined by "if one 1 is present the result is 1, else it is 0". > > > This covers also the infinite case. > > > > It does not. Define: > > A1 = 0.10... > > A2 = 0.110... > > etc. Now I can do: > > SUM{i = 1 .. n} A_i > > where SUM means repeated application of *+. There is nothing in your > > definition that covers the infinite case. > > If any case includes at least one 1 then the *+ sum is 1. Hence there > is no necessity at all to distinguish cases. Yes, it was sloppy terminology. What happens when n grows without bound? What is lim{n -> oo} SUM{i = 1 .. n} A_i ? How do you define that? Without such a definition I have no idea what the result is when I *+ all An. > > > But it is a unary representation of a whole number, namely of the > > > number of digits of a real number. Nevertheless, in decimal > > > number of digits of a real number. Nevertheless, in decimal > > > representation 0.111... does exist, but not in the list. The addition > > > of the list numbers gives the largest number of the list. > > > > In the finite case. As you have not provided a way to do the infinite > > sum, I have no idea what happens in that case. > > Define: If any case includes at least one 1 the the *+ sum is 1. Again, in the finite case. You have not defined what you mean with the infinite sum. I am not going to *+ "infinitely many" times. I have other things to do. > > > Whatever this > > > may be, it is not 0.111... . Proof: *Every* list number ends up with at > > > least one zero. Therefore there must be at least one column with only > > > zeros. > > > > Wrong. Provide a proof of that statement if you think it is correct. > > What is wrong? The statement that every number in the list ends by 0? No, the statement that there is at least one column with only zeros. Pray provide a proof. > > > > Makes no sense to me. I did state: > > > > For all p there is an n such that An[p] = K[p]. > > > > You said that was wrong. I asked you why that was wrong, and you come > > > > up with some nonsense. I state: select n = p. Isn't it a fact that > > > > For all n, An[n] = K[n]? > > > > > > All n which can be indexed are in the list. K is not in the list. > > > > This is not an answer to my question. What is wrong with > > For all n, An[n] = K[n]? > > Pray, is there an n such that An[n] != K[n]? If so, what n is it? > > It is not such an n. So the statement For all n, An[n] = K[n] is true? As is the statement For all p there is an n such that An[p] = K[p] also true? Why than do you write that it is false? > But if K would consist of such sequences only, > then it was in the list. Still not given a proof. > > > All segments K[n] are in the list. K is not in the list. What > > > distinguishes K from all its segments if not at least one more 1at a > > > position which cannot be indexed? K has nothing else to offer. > > > > What K offers is that it has no zeroes in its representation. And all > > its digit positions are natural numbers. > > Then all its digit positions are in sequences in the list, because all > natural numbers are in the list. Right. > This would mean K is in the list. Wrong. -- 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 16 Jul 2006 19:57 In article <1153053505.808769.75210(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > So again, what is the definition of "transformation"? > > > > > > The definition of *allowed transformations* was given by Cantor in > > > above text. There is *no* further question except that you are > > > unwilling to confess your error and your insult that I was unable to > > > understand the German text. > > > > Yes. But I do not know whether all changes to the ordering are > > "transformations". So how is "transformations" defined? > > In German it is "Umformungen", that means "change of the order". Cantor > says: those changes, which and only which can be traced back to > finitely many or infinitely many transpositions of elements, each > transposition including two elements. So you claim that Cantor meant unrestricted re-ordering. If he meant that, his statement is false. That is entirely possible, of course. > > > It is not the question whether Cantor was wrong, but only what was the > > > meaning of his quote. > > > > Sorry, you use the quote as proof of something. In order to do that you > > need to know whether the Cantor was wrong or not. > > A transformation does not disturb a well-order. > If infinite sets do exist and can be exhausted, then the set of my > transpositions does exist and can be exhausted. Now you use an entirely new term: "can be exhausted". I think you mean that you can take out elements one by one and doing this at some stage the infinite set becomes empty. Howver, I think, that if that can be done, that there is a last element you can take out. And, according to the axiom of infinity, that is not possible, so infinite sets can not be exhausted in this sense. > That's all which is necessary to disprove set theory. You use a contradiction of the axiom of infinity (can be exhausted), so there is no disproof. Now try the same with the assumptions that infinite sets do exist and can not be exhausted. > > > > > Yes, that is a subset of the Umformungen. What is his definition of that > > > > term? What does he mean when he writes "Umformungen"? > > > > > > I you are you unable to read English as well as German, try to get a > > > copy in Dutch. > > > > Pray complete for me: transformations are ... > > Only "such transformations" need be defined. Such transformations are > changes of the order which can be accomplished by transpositions > (interchanges) of two elements. Well, I think *any* re-ordering can be accomplished by a sequence of transpositions of two elements. But when I write: The question, though which froobles the ordinal number of a well-ordered set is changed, and through with it is not changed, is easily answered, those, and only those froobles leave the ordinal number unchanged that can be rewritten as a finite or infinite set of transpositions, that is, of interchanges of two elements. you do not need the definition of froobles? I can tell you that my statement is entirely correct. (I can state that your sequence of transpositions does not form a frooble.) > > > There is no proof necessary. If infinite sets exist and can be > > > exhausted, then the set of transpositions can be finished. As no single > > > transposition changes the ordinal number, it remains unchanged during > > > the whole proess.. > > > > Ah, but you have here an additional qualifier: "can be exhausted". Well, > > infinite exists (the axiom of infinity), but can not be exhausted with > > extraction of one element each time. So now what? You need a proof. > > If they cannot be exhausted, then they do not exist actually but only > potentially. Then they have no cardinality. Eh? Infinite sets do exist, but they can not be exhausted. So why do they not have cardinality? It is entirely possible to define cardinality for them. What is your problem with that? > > In that case Cantor was wrong. > > Reorder the naturals to (1, 2, 3, ..., 0) (yes, Bourbaki), and see that > > the order type is now w+1, so it did change from w. On the other hand > > that re-ordering can be re-written as an infinite sequence of > > transpositions. > > One should have seen that earlier, then Borbaki would not have > succedeed to define 0 as natural number, even in political decisions. Pray explain the last part "even in political decisions". But I can rewrite it in non-Bourbaki: Reorder the naturals to (2, 3, 4, ..., 1) and see that the order type is now w+1, so it did change from w. > > > You will have to become accustomed with that. As only potential > > > infinity exists, not infinite process can be regarded as finished and > > > no sequence of 1's can be regarded as static unless it is finite. > > > > And this again makes absolutely no sense to me. In the definition of > > 0.111... there is no infinite process involved. > > If the 1's cannot be exhausted, then there is only a process to > consider one after the other without comng to an end. You are fighting the mathematical use of limits. On what basis do you allow the computational use? -- 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 17 Jul 2006 09:29 Virgil schrieb: 0.0... > > > > That is not a natural number. > > It is to John von Neumann and he know more about it than you do. If Newton had eggs called apples, that would have been just as wrong as the error of John von Neumann. > > > > > > > > So let's see: > > > {0.0...} has cardinality {0.10...} > > > {0.0..., 0.10...} has cardinality {0.110...} > > > > {0.10...} is not a cardinality but a set. > > In NBG cardinalities are sets. In Transsylvania horses have four heads and only one leg. The cardinal number remains if the type of elements and their order in a set is neglected. > > > > By indexing we see {0.10...} has cardinality 0.10... > > {0.10..., 0.110...} has cardinality 0.11... > > Not in NBG. To my knowledge v. Neumann and Bernays and Gödel agreed that their cardinality was 3 --- and not 4. Regards, WM
From: mueckenh on 17 Jul 2006 09:34
Virgil schrieb: > > Obvious is that no transposition of two elements of a well-ordered set > > can destroy the well-order. > > Obvious is that, if infinite sets do exist and can be exhausted, the > > set of my transpositions does exist and can be exhausted. > > > The issue is what would result. > > Since at any finite stage one still has infinitely many infinitely > descending sequences of rationals by magnitude, what evidence is there > that "exhaustion" would produce anything else? Because "exhaustion" is defined by reaching the limit, the order by size. Unless it is reached, numbers need be transposed, and hence, the set of necessary transpositions is not yet exhausted. Regards, WM |