An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Dik T. Winter on 12 Jul 2006 19:50 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 Pl?tze in der gesamten Rangordnung wechseln, so wird dadurch > der Typus nicht ver?ndert, also auch nicht die "Anzahl" oder > "Ordnungszahl". Daraus folgt, da? solche Umformungen einer > wohlgeordneten Menge die Anzahl derselben unge?ndert lassen, welche > sich auf eine endliche oder unendliche Folge von Transpositionen von je > zwei Elementen zur?ckf?hren lassen, d. h. alle solche ?nderungen, > 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. 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? 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. -- 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 12 Jul 2006 20:45 In article <1152736018.461492.108050(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > > Indeed, at that time it was not yet an axiom, but the thought was > > > > present that infinite sets did exist. > > > > > > There was not such an assumption in general. Infinite sets did not > > > exist (as they do not exist yet), but only the potential infinite was > > > accepted. Cantor was little understood and was blamed to do philosophy > > > or theology but not mathematics. > > > > "... die Gesamtheit aller endlichen Zahlen 1, 2, 3, ..., v, ...", Cantor. > > just because of views like that. Yes, so what? As I wrote, "it was not yet an axiom, but the thought was present that infinite sets did exist". And indeed, Cantor had not yet formulated it as an axiom, but nevertheless did state it. What you are arguing against is *not* against the proof, but against the axiom. > > > It leads to the inconsistency that you must demand 0.111... *- (0.1 + > > > 0.11 + 0.111 + ...) = 0.000... but 0.111... not being in the sum. > > > > As you have not defined the meaning of the second term, this makes no > > sense. Offhand I would say it goes 0.1, 0.21, 0.321, 0.4321, 0.54321, > > 0.654321, 0.7654321, 0.87654321, 0.987654321, 1.0987654321, and I see > > no obvious limit emerging when we do the infinite sum. > > The numbers are to be understood as sequences or vectors, so your last > sum is 0.(10),9,8,7,6,5,4,3,2,1. The next would be 0. > (11),(10),9,8,7,6,5,4,3,2,1 and so on. The magnitudes of the numbers do > not matter if they are larger than zero, because 1 *- omega = 1 *- 1 = > 0. My question is whether > > 0.111... *- (0.1 + 0.11 + 0.111 + ...) = 0.000...? Next question. What is the meaning of "+ ..."? How do you define that? I remind you that you are trying to show an inconsistency with the axiom of infinity, so you should define it using that axiom in order to be able to show an inconsistency. And so there is no largest natural number. > > > And you must > > > refuse to calculate with fractions in mappings. > > > > Here I have no idea what this means. > > The binary tree: Your binary tree again. What is the relation with "refuse to calculate with franction in mappings"? Why do you never give direct answers to questions? -- 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 12 Jul 2006 20:54 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? -- 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 12 Jul 2006 22:29 In article <1152735810.530270.91150(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > The representation 0.111... *does* exist. There is a specific definition > > for it (as I have already given), and an axiom through which you may > > prove that it does exist. > > 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. *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. > > > What is wrong about it? Take some particular p and take n any value larger > > than or equal to p. We see that An[p] = 1 = K[p]. So what is wrong? > > If you take some particular p, this p is a natural, otherwise you could > not take it. Therefore you cannot show that there are digits in > 0.111... which are not indexed by naturals. 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? > 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. > > > 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. > > > > All digits which are indexed by smaller list numbers can be indexed by > > > one larger list number. Therefore always only one is required. > > > > Care to explain the implication you mention above? > > How does K differ from all natural unary numbers together? 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 do not need the first two list numbers 0.1 and 0.11, because > > > > > the third alone is sufficient: 0.111 does index the first three > > > > > digits p = 1 to 3 of 0.111... > > > > > > > > I am a bit at a loss here what you mean with "indexing". > > > > Again care to explain? How can a single number index three digits? > > instead of "to indeex" you can also say "to cover", i.e. to have more > 1's than (or at least as many as) the number indexed or covered. Well, to index can be stated as to cover. But not the other way around. > > > So let us find a resolution of this dilemma by mathematics. > > > > > > Define "*-" by > > > a_i *- b_i = a_i - b_i if a_i > b_i else a_i *- b_i = 0 > > > (subtraction down to zero but without negative numbers) > > > > > > Consider 0.111... and the list numbers as sequences or as vectors, such > > > that *- can be applied to each term separately. > > > > > > If you are right, you must maintain the result: > > > > > > 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 "+ ..."? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 13 Jul 2006 04:35
In article <1152638895.480011.23630(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > But not all unary sequences are representations of natural numbers. > > 0,111... is a sequence which does not represent a natural. > > And it cannot be completely indexed by natural numbers. That is precisely wrong, it is only the set of all natural numbers which CAN index it. |