An uncountable countable set
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From: mueckenh on 13 Jul 2006 06:42 Dik T. Winter schrieb: > In article <1152735445.836339.213400(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > The first well-order taken from Cantor's. The others remain well-orders > > because: wenn in einer wohlgeordneten Menge irgend zwei Elemente m und > > m' ihre Plätze in der gesamten Rangordnung wechseln, so wird dadurch > > der Typus nicht verändert, also auch nicht die "Anzahl" oder > > "Ordnungszahl". Daraus folgt, daß solche Umformungen einer > > wohlgeordneten Menge die Anzahl derselben ungeändert lassen, welche > > sich auf eine endliche oder unendliche Folge von Transpositionen von je > > zwei Elementen zurückführen lassen, d. h. alle solche Änderungen, > > welche durch Permutation der Elemente entstehen. > > I translate (not needed for Franziska, but needed for others): > When in a well-ordered set any two elements m and m' change place in > the common order, the order-type will not change, so also the ordinal > number will not change. From that it follows that those transformations > of a well-ordered set leave the ordinal number unchanged, that can be > written as a finite or infinite sequence of transpositions of two > elements, that is, all such changes that emerge through permutations > of elements. > > I see a problem here: > > It is not clear what the meaning is of the words "transformations" > ("Umformungen") and "permutations" ("Permutation"). I think, it is clear > that it does not include *any* re-ordering, although many of them can be > written as an infinite sequence of transpositions, while some those *do* > change the ordinal number. These words are uninteresting. Transpositions or exchanges of two elements are clearly defined. Infinitely many are admitted by Cantor. > > The reordering of the naturals (0, 1, 2, ...) to (1, 2, 3, ..., 0) can > be written as an infinite sequence of transpositions, but it changes the > ordinal number. But is it a permutation? Unimportant for Cantor's statement. There is no transposition which could change the ordinal number. Only the infinite process does what cannot be done by any finite set of transpositions. This shows that infinite processes and infinity is doubtful. My example with |Q shows that infinity is impossible. > > On the other hand, there are many infinite sequences of transpositions > that do not change the ordinal number. Consider a well-ordered set of > type w * w. Interchange the first two elements of each maximal subset of > order type w. > > Anyhow, either WM's interpretation is wrong, and so his conclusion is wrong, > or WM's interpretation is right and Cantor's statement is wrong, and so > (again) WM's conclusion is wrong. > > I think the interpretation by WM is wrong. I know your interpretation is wrong. But that does not at all affect my proof, because, as Virgil, emphasized frequently, there is no transposition which changes the well-order to normal order while maintaining the initial enumeration (= well-order). Would infinitely many transpositions be possible, this would necessarily happen. Regards, WM
From: mueckenh on 13 Jul 2006 06:48 Dik T. Winter schrieb: > In article <1152736417.397578.308650(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > > > I proved in my special list even that the diagonal number is a > > > > > > rational. > > > > > > > > > > I wonder whether it was a proof or just some handwaving. > > > > > > > > 0.0 > > > > 0.1 > > > > 0.11 > > > > 0.111 > > > > ... > > > > replace 0 by 1. > > > > > > As far as I see the diagonal starts with 1.000... Am I right? > > > > I use only the digits behind the point: So the diagonal is 0.111... = > > 1/9. > > So you imply additional 0's after your notation. I was not sure. > > > If there is no other outcome possible, I don't need a further > > definition. 0.999... = 1 follows from the definition of (+,-,*,/) in > > the real numbers. > > Pray tell me how. Somewhere else you stated that the representation > 0.111... did not exist. What are you arguing? Either the representation > 0.111... does exist or not. And if it does exist the definitions of the > mathematical operations are not sufficient to give a meaning to it. > > Anyhow, how do you show that 1.000... - 0.999... = 0 with the definitions > you are using? In "usual" mathematics we have 10 * 0.999... = 9.999... and from that we get easily 0.999... = 1. I argue that 0.111... does not exist. When I will have succeded, also 0.999... will be abolished. But that has not yet been generally accepted. OK? Regards, WM
From: mueckenh on 13 Jul 2006 07:03 Dik T. Winter schrieb: > > The axiom says "there is an infinite set". It does not say that > > 0.111... does belong to that set, in particular because all numbers > > which in fact do belong to the set are different from 0.111... . > > You are using the word set with two different meanings in the same sentence. The set of naturals shold be the same set as the set of indices of 0.111... . But it cannot. > *And* you do not answer my primary comment. There is a specific definition > for a notation like 0.111... Without such a definition it is just a string > of eight symbols. There is a specific definition that the set of all naturals has a cardinal number larger than any element of the set. This definiion is wrong. I show this by the fact that 0.111... is not in the set of unary reprsetations of all natural numbers. aleph_0, the number of 1's in 0.111... is *not* the cardinal number of |N. > Again, not an answer to a specific question. I wrote: > > For all p there is an n such that An[p] = K[p]; > and you wrote that is wrong. What is wrong about that statement? Te assumption that all digits of 0.111... can be inexed by natural numbers while 0.111... has the property to to be outside of the list. > > > But as K is not in the list > > of natural numbers, it must differ from the numbers in the list. And > > this difference cannot be accomplished other than by K having more > > digits than any number in the list. Again, this is impossible. Hence we > > have a contradiction. > > Pray *prove* why that is impossible. You always only state it but provide > not prove. Each An has finitely many digits. K has infinitely many digits. > If you disagree with the second you disagree with the axiom of infinity, > but that is philosophical. K cannot have infinitely many digits if all can be indexed by finite numbers. Infinite is larger than any finite number. Therefore every list number taken will not be capably of indexing K. 0.111... - 0.111...1 = 0.000...0111... > > > > > > > > such that there are more 1's in K than in any An. > > > > > > The last is true. But it is also true that for all p there is an An. > > > Or please exhibit a p for which it is not wrong. > > > > This equation > > 0.111... - 0.111...1 = 0.000...0111... > > says that for any An = 0.111...1 there is a p = n+1 which cannot be > > indexed. > > This makes absolutely no ense to me. And again is no answer to the > question I posed. Whatever p you choose, there is always a place in 0.111... which cannot be indexed. Therefore there is a place which cannot be indexed in principle. > Again you fail to answer a question, Your implication was: > > > > If you think the sentence "all positions of K = 0.111...are indexed by > > > > list numbers" is not equivalent to the sentence "K > > > > is in the list", then you seem to imply that more than one list number > > > > is required to index the digits of K. > Care to explain it? You know that one list number is not capable of indexing 0.111... , bu you assume that 0.111... can be indexed completely. Therefore more than one list number should be required. But this would not help. > Well, to index can be stated as to cover. But not the other way around. Therefore never more than one number is required for indexing purposes. > > > > > > > > > 0.111... *- (0.1 + 0.11 + 0.111 + ...) = 0.000... > > > > > > I can make no sense of the second term. What do you *mean* with > > > (0.1 + 0.11 + 0.111 + ...) > > > > The sum of all list numbers by digit, as shown in the following > > example: > > > > 0.1 > > +0.11 > > +0.111 > > _____ > > =0.321 > > Ok. What about "+ ..."? Definition: 1+1+1+... := 1 Regards, WM
From: mueckenh on 13 Jul 2006 07:31 Virgil schrieb: > >That the transpositions are > > conditional does not disturb the proof. > > It does mean that they must be performed in order though. > > And that does distrub the proof. Why should it? Do you dislike order? Cantor's list is ordered, the set N is ordered. Do they not exist for this sake? is there any hint why order should destroy infinity? > > > Cantor's diagonal argument is > > also conditional. > > But there are no conditions on any digit which depend on any other digit > having been calculated previously, i.e., no sequential conditions. Thus > one can have a general rule that does eachdigit without reference to any > others. so is valid for all in one rule and one step. In order to identify any digit, you must find it. That requires counting. That requires order. > > > That the conditions have to be executed one after the > > other does not disturb the proof. > > It means that the"mueckenh" process, as described, can never end. > This is a set and it exists as well as any other infinite set. Not more and not less. > > > > Cantor's diagonal argument requires > > counting which is a sequential act too. > > Not at all. Did you ever count? Or do you mathematics without numbers but with letters only? > Getting the list in the first place may require counting, > though that is dubious, but once it is presented, no further counting is > required. Finding the n-th line of the list, and within this line the n-th digit requires hard counting, if n is large. Regards, WM
From: mueckenh on 13 Jul 2006 07:35
Virgil schrieb: > In article <1152735810.530270.91150(a)p79g2000cwp.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > The axiom says "there is an infinite set". It does not say that > > 0.111... does belong to that set, in particular because all numbers > > which in fact do belong to the set are different from 0.111... . > > > But 0.111... in effect IS that set to which it does not belong. Yes. 111... represents aleph_0 or omega. But we see that it is impossible to have aleph_0 or omega natural numbers. Every smaller number is finite. If aleph_0 is the first transfinite cardinal, then the set of natural numbers alone is not actually infinite. Regards, WM |