An uncountable countable set
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From: Franziska Neugebauer on 16 Jul 2006 08:32 mueckenh(a)rz.fh-augsburg.de wrote: > > 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. Do you want to raise an objection? >> >> 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, non sequitur. > or there must be one number having 1's in every column but no zero at > all. Not _in_ _the_ _list_. Also non sequitur. You may recall that every sequence member has trailing zeros. > 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. > 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... . 0,111... is not "in the list" by definition of the list _and_ the *-sum of the list still is 0.111... This is neither inconsistent nor contradictory. > You claim a third alternative which is prohibited by tertium non > datur. My "list of claims" is finite and already finished. There is no third alternative. > 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. I am not going to waste my precious time on scrutinizing this futile mess. F. N. -- xyz
From: Franziska Neugebauer on 16 Jul 2006 08:34 mueckenh(a)rz.fh-augsburg.de wrote: > > 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. There is no contradiction. For all j e N there is an n e N such that a_jn = 1 namely n = j, since a_jj = 1. F. N. -- xyz
From: mueckenh on 16 Jul 2006 08:38 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. > > > > 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. 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. That's all which is necessary to disprove set theory. > > > > 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. > > > > > 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. If they cannot be exhausted, then they do not exist actually but only potentially. Then they have no cardinality. > > > > 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? Yes. But not every transformation must be a re-ordering. > 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. > > > > > > > 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. > -- If the 1's cannot be exhausted, then there is only a process to consider one after the other without comng to an end. Regards, WM
From: David Hartley on 16 Jul 2006 09:59 In message <1153051856.174981.269510(a)35g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de writes > >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? >> >> Answer: Those which are realised by a finite or infinite set of >> transpositions. >> >> "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. >In German we read "Umordnung" which even better would be translated by >"changing the order". And this can be done by transposing two elements >infinitely often. >By the way, why should Cantor have written that text, if it did not >express something? I'm sure it meant something. If his definition of "Transformation" includes all sequences of transpositions, and his definition of the effect of such a sequence is the same as mine (and, I assume, as yours, although you have never stated it) then it has a clear meaning, but is easily disproved (in ZF). If his definition of "Transformation" is different then his claim may be correct. But that is of no use to you until you can show that your sequence of transpositions is a transformation in his sense. And even then, you still need a *proof* of Cantor's claim. Far better to forget about Cantor and provide your own proofs. -- David Hartley
From: mueckenh on 16 Jul 2006 12:15
David Hartley 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. > >That's all needed. > >Belief in anything else is matheology. > > Obvious is, you haven't the faintest idea of how to prove your claims. Obvious is that the finite or infinite binary tree cannot contain more paths than edges. | o /\ Obvious is that the addition *+ changes the sum of the list 0.1 0.11 0.111 .... 0.111...1 whenever another line, not yet included, is added. But it does not change the sum of 0.1 0.11 0.111 .... when the line 0.111..., not yet included, is added. No suspicion raised? Faithful as ever? Regards, WM |