An uncountable countable set
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From: Dik T. Winter on 16 Jul 2006 07:06 In article <1153040519.966908.158530(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > Defined is Umformungen: "diejenigen und nur diejenigen Umformungen = > > > those and only those transformations" > > > > Yes, so that is not a definition of transformations, it restricts the > > transformations to a subset of them, namely only those transformations > > that have a particular property. > > Of course. "Umformungen" includes many transformations like those of > iron to hammers. Yup, and may not include some other changes. > > Apparently there are transformations > > that do *not* have that property. So the question remains, what is the > > *definition* of "transformations"? (I can easily enough give a > > transformation that does not have that property at all: transform {1, 2} > > to {1} by mapping both elements of the first to 1. It can not be > > re-written as a sequence or set of transpositions. So while it is a > > transformation, it does not satisfy the requirements to leave the > > ordinal number unchanged.) > > > > 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? > > If it is > > unrestricted (and so any re-ordering is allowed), the statement is > > trivially false. > > 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. > > If there are restrictions (and so not any re-ordering > > is allowed), the statement might be true. But whatever, when the > > first is true (any re-ordering is allowed as a transformation), the > > statement is false and you can not use it, because it is not proven. > > On the other hand, if there are restrictions, you have to show that > > your set of transpositions form a transformation before you can use > > the statement. > > > > > they are defined by: "welche sich zur=FCckf=FChren lassen auf eine > > > endliche oder unendliche Menge von Transpositionen, d. h. von > > > Vertauschungen je zweier Elemente =3D which can be traced back to a > > > finite or infinite set of transpositions, i.e., interchanges of two > > > elements." > > > > Yes, that is the definition of the property of the transformations that > > leave the ordinal number unchanged. But the question remains, given > > any sequence (or set) of transpositions. Is the result a "transformation"? > > I do not know, because I do not know the definition of "transformation". > > Then read again the text in English which you quoted above. There it is > stated explicitly. Pray do also reread. There is *no* definition of "transformations". > > > He says it clearly: those and only those transformations which are > > > realized by a finite or infinite set of transpositions, i.e., > > > interchanges of two elements, do not change the Anzahl =3D ordinal > > > number. > > > > 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 ... > > > No. Cantor obviously was correct, if infinite sets would exist. > > > > Again you state "obviously" when I ask for a proof. So you are apparently > > not able to provide a proof but think that what Cantor writes is always > > true. Who coined the term matheology? > > 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. > > See above where I have written my rebuttal. You apparently have no idea > > what a good definition entails. He does not define the term "Umformungen", > > but defines a subset of "Umformungen" that have some particular property. > > Until we know that the larger term "Umformungen" means, we have no idea > > what that subset contains. > > The full meaning is uninteresting, as it covers even mechanical > transformations. For our case only those tranformations are > interesting, which Cantor defined precisiely. So every re-ordering is a transformation? 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 > > > > > Therefore it never becomes 0 but always remains > 0 like 0.11... > > > > > never becomes 1/9. > > > > > > > > But apparently you do not use the standard definition of 0.111... . Is it > > > > a number or something else? > > > > > > It is a number with aleph_0 places, which are all occupied by 1's. > > > > Eh, it is a number that changes value? You state "never becomes". Strange > > to state something like that about a number. > > 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. -- 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 16 Jul 2006 07:47 Franziska Neugebauer schrieb: > Every column contains at least one 1. After the definition of the > columnwise infinite *-summation (yours and mine) every column thus > is *-summed up yielding 1. Every column? You said that every natural differs from 0.111... by the fact that it has trailing zeros. > > >> So it is not valid. If you drop it, the contradiction vanishes. > > > > An addition cannot supply 1's at all places if all summands end with a > > 10, i.e., one,zero combination. > > There is no column not containing a 1. > Then there must be either numbers which have zeros distributed like the following example: 0.111000111... and so on, or there must be one number having 1's in every column but no zero at all. 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. Hence (which means the same as "therefore, for this reason, thus, that is why, ...") there are only two logical alternatives: Either 0.111... is in the list, or the *+ sum of the list numbers is not 0.111... . You claim a third alternative which is prohibited by tertium non datur. This can be proven in another way too: 1) If you remove all numbers with more than n 1's, then every line of the list contains at least one zero. Mixed numbers are not present, hence (which means the same as "therefore, for this reason, thus, that is why, ...") the zeros mount up at the right hand side. There must be a column with zeros. And there are only n columns which give the *+ sum 1. 2) If you reintroduce a number larger than n, then the number of columns with 1's grows. 3) If you reintroduce all numbers with a finite natural number of 1's, then the number of columns grows further, but there are only a finite natural number of columns with 1's. You said all these numbers carry trailing zeros. But now you refrain from this all-quantifiyer which according to he definition of *+ sum would yield a sum zero, but assert that the 1's of these numbers cover all columns. 4) If you at last introduce 0.111... in the list (as the last line or as the diagonal number, each of its 1's in a separate line), then the *+ sum remains he same. As 0.111... carries more 1's than all numbers in the list (i.e., not less than and not equal to any other) it is odd that its introduction should not have an effect. Concluding, you can assert further, that all these deductions are in vain because infinity could not be refuted as little as the existence of God can be refuted. And if you refrain from logical deductions you are right. But if infinity were a number and would behave like a number, it could be refuted. Hence your improvable infinity is completely outside of mathematics, it is not a number and is not good for anything else. Regards, WM
From: mueckenh on 16 Jul 2006 07:49 Franziska Neugebauer schrieb: > Once again: There is no column without a 1. /Hence/ every column *-sums > up to 1. Contradiction to A n : n has trailing zeros. Regards, WM
From: mueckenh on 16 Jul 2006 07:54 Virgil schrieb: > In order to justify the "0." part of "0.1" 0r "0.11" , etc., > one must have a natural number base greater than 1, otherwise the "0." > is redundant. "0." := "Look, here comes a unary representation of a whole number." > > > > > > > > The result is the same. Hence, if 0.111... exists, > > > > it is in he list and is not in the list. > > It exists but not as any of its predecessors. > > > How then can it be the *+ sum over its predecessors? Regards, WM
From: mueckenh on 16 Jul 2006 07:57
David Hartley schrieb: > >> Can you provide a proof of that statement? Because you state just above: > >> "Cantor's conclusion is correct, with no doubt, if infinite sets like > >> the set of my transpositions do exist." Which contradicts what you state > >> here. > > > >No. Cantor obviously was correct, if infinite sets would exist. But > >they do not. So he was wrong. And you are wrong, as I have shown above > >by Cantor's text which you misinterpreted grossly. > > Cantor's proposition (as you interpret it) leads directly to a > contradiction in standard set theory. So it is disproved. To show that > set theory is inconsistent you must also show that it can be proved. You > cannot do this by arguing over quotations from Cantor. He does not > appear to have offered a proof and he was not infallible. You claim it > is obvious, so prove it yourself, with no more quotations, (except > references to standard theorems}. 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. That's all needed. Belief in anything else is matheology. Regards, WM |