Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Dik T. Winter on 2 Nov 2006 08:44 In article <1162405779.057915.322020(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > Therefore it is impossible to exchange omega letters in a diagonal. > > > > Wrong. For each element of the list a digit is calculated in the diagonal. > > As there are infinitely many (omega) elements in the list, there are > > infinitely many (omega) digits in the diagonal. > > omega is the supremum, not the maximum. It does not contribute a > diagonal digit. And it does not contribute a line. But the number of lines is omega. And the number of digits is omega. > An infinite diagonal requires not only an infinite length but also an > infinite width of the matrix. Therefore your absurd infinite number of > finite lines does not help you. Here we have the same facts as in our > old problem > 0.1 > 0.11 > 0.111 > ... > > you remember? Without an infinite number in the list there is no > infinite diagonal defined. I state (you know that) that the diafonal: 0.111... > > > The > > > diaogonal cannot be roader than the list. The length of the diagonal is > > > the minimum of width and length. This knowledge is prior to your > > > axioms. > > > > Width and length are equal. > > Fine. But the width is finite by definition. (We do not put finite > segments together, but we have only finite seqments.) Nope. The width is unlimited, as is the length. And so is not finite. > > > A matrix with width A and length B has a diagonal which has min(A,B) > > > elements. If your axiom contradicts this, then the axiom contradicts > > > mathematics and should be abolished. > > > > That is not contradicted. Width and length are equal. > > This amounts to say that there are infinite natural numbers or that the > diagonal is longer than any line. Wrong. > The maximum of a set of finite numbers which has no maximum is simply > not present. Right. > It is *not* an infinite number which is larger than any > finite number, because a maximum must belong to the set. Right. > And a supremum > not belonging to the elements of the set does not yield a diagonal > digit. And again right. And still your conclusion is wrong. Obviously the diagonal has no finite length, because if it had it would have the same length as one of the numbers, and then the next number would be longer, which is a contradiction. So the diagonal is infinite in length. > > > And if you can conclude that in this context every straight line > > > crosses itself 17 times, then you will also take that as a fact? > > > > You, if that follows from some axiom, it would really be possible, unless > > the added axiom leads to an inconsistencey. But I think there might be > > surfaces where that is even valid. > > Let us stick to Euclidean geometry. But that is unimportant. I see it > is impossible to convince you of the existence of reality. What is the "existence of reality" in this context? -- 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 2 Nov 2006 09:35 In article <1162406116.713481.142930(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1162300936.776151.45540(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: .... > > > > You keep stating that, without proof. That it has only finite initial > > > > segments, I agree. Not the remainder. > > > > > > It is you who denies proof. Give an example of a digit which does not > > > belong to a finite sequence. If you cannot do so, then every digit > > > belongs to a finite sequence. > > > > I do *not* deny that. But there are infinitely many digits, each being > > part of a finite segment. > > That is but a statement without value. If you do not deny my assertion, > then it is not necessary and not useful to assume any infinity, because > every number we can name belongs to a finite set. It is indeed not necessary, just reject the axiom of infinity, and try to develop mathematics without it. *With* the axiom of infinity, there *are* infinite sets. > > > The sequence 0.111... consists of every > > > digit but not of more. Therefore, there is not need and no use of > > > talking about infinite sequences. > > > > How can those infinitely many digits form a finite sequence? > > There are not infinitely many digits. That is only your illusion. Indeed, still only rejecting the axiom of infinity. And at the same time telling that you are working within the context of the axiom of infinity. > > No, something quite different was argued there. Namely that the limit > > *also* is the number of edges in the infinite tree (or somesuch) > > requires transfinite induction. > > But is does not. Indeed, it does not. But I thought you were maintaining that it would be? > > > Induction will show everything that *can* be shown. Up to every finite > > > position 0.111...1 is rational. More is not possible. Irrational > > > numbers don't exist, but that is another topic. > > > > Yes, you again deny that the infinite exists. Again denying the axiom of > > infinity. While still maintaining that you are arguing within the > > axiom of infinity. With the axiom of infinity it exists, and irrational > > numbers exist, 0.111... does exist, and induction can not show that > > 0.111... is rational. > > > > > Anyhow, in the binary tree there is no transfinity required. > > > > There is. Namely to show that the number of edges in the infinite tree > > is equal to the limit of the number of edges in the finite subtrees. > > There is no omega required in the definition of the real numbers, but > transfinite induction? No. I have no idea how you come at that idea. > There is transfinite induction required to see that the set of digits > of a real number has cardinality aleph_0. No. I have no idea how you come at that idea. > There is transfinite induction required to see that LIM 1/n = 0 and > that 1/9 = 0.111... ? No. I have no idea how you come at that idea. > Where is the transfinite step in Cantor's list, showing that the limit > of the digits of the diagonal number is a real number? None is needed. I have no idea how you come at that idea. > And that this > real number differs from every list entry also in this limit? Just this > latter assumption being wrong. You need transfinite induction to show that "the number of edges in the infinite tree" is equal to the limit of "the number of edges in the finite tree". > > > And it exists in many other disguising. But it does not exist as a > > > number which can be put in trichotomy with all other numbers. > > > Why not? Anyhow, I presume that you use linear algebra on occasion and > > eigenvalues of matrices. How can you use things that do not exist? > > You should know it, in particular if talking about transfinity and > transfinite induction. > But to answer your question: I can use the idea. I can use the > undisputed fact that the square of this idea is 2. And mathematicians call it a number. So what? > > Sorry, that is not a valid HTML document and will not display properly. > > It is mhtml. It will not display properly with my browser (Mozilla). And using Internet Explorer is no option (it is not supported on the machines I have access to). > > I think it is excluded in your finitistic world. And, I think, that > > indeed the universe is not able to provide the computer power to actually > > observe such a sign change. To do so would mean to actually calculate > > pi(x) for x about 10^316. The best current method has a time complexity > > of O(x/(ln x)^3) and a storage complexity of O(x^(1/2)/(ln x)). > > Please calculate that in your infinitistic world - if you can. You want to calculate such things in your finistic world. In the infinitistic world such calculation is not necessary; an existence proof is sufficient. -- 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 2 Nov 2006 10:46 In article <1162470378.676172.129570(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1162405520.008395.100850(a)e64g2000cwd.googlegroups.com> muecke= > nh(a)rz.fh-augsburg.de writes: .... > > > > > > And in mathematics 1/oo is *not* defined. > > > > > > > > > > Not in mathematics. But in a theory which assumes omega to be > > > > > a whole number. > > > > > > > > Which theory? > > > > > > Set theory. > > > Cantor invented omega and defined omega as a whole number. > > > Who changed this standard meaning? > > > Why do you think this meaning was changed? > > > When do you think the contrary meaning became standard? > > > What is the contrary meaning? > > > Do you agree that A n: n < omega is incorrect? > > > If not, why do you complain about on-standard meaning on Cantor's > > > definition? > > > > A nice rant. > > Why don't you answer? Because it is unrelated with the question where 1/oo was defined. But to answer, I do not know who and when the definition of ordinal number was introduced. I know *why* it might have been introduced: to remove possible confusion. I have no idea what you mean with "contrary meaning". As your statement A n: n < omega does not indicate where n is coming from, I can not state whether it is correct or not. But if the natural numbers are meant, than it is certainly correct (if we assume the natural comparison between ordinal numbers and natural numbers). > > > Where did Cantor define 1/oo? A quote might be > > appropriate. > > see below .... > > Why? As long as 1/omega is not defined you can not talk about it. You > > simply assume that 1/omega is a number. But that is not the case, it > > is not defined. > > If omega is a number > n, then 1/omega is a number < 1/n. For all n e > N. First, how do you *define* 1/omega? You know how arithmetic normally is defined? But of course, once you define it, some properties of arithmetic are lost. What is omega * (1/omega)? > > > In particular the limit of all segments of N is not defined. But set > > > theory does it. > > > > No. And if so, show me where. > > Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu > denken, welcher die Zahlen n zustreben, wenn darunter nichts anderes > verstanden wird, als da? omega die erste ganze Zahl sein soll, welche > auf alle Zahlen n folgt, d. h. gr??er zu nennen ist als jede der > Zahlen n. (p. 195) > > This definition has been conserved up to our days: Limesordinalzahl or > limit ordinal number. Yes. But "limit" has not really been defined. > > > Everything is a set (in ZF). Numbers are sets too. > > > > In the Von Neumann model of ZF, I would think. > > In ZF. > > Karel Hrbacek and Thomas Jech: "Introduction to set theory" Marcel > Dekker Inc., New York, 1984, 2nd edition, p. 2: "So the only objects > with which we are concerned from now on are sets." Again, depends on the model. But even if we allow that, it can not be applied to the ball and vase problem. In that case there is a limit involved (in your models of that problem). And to use those limits (as you do) you have (1) to define the limit of a particular sequence of sets (2) to show that the cardinality of that limit is equal to the limit of the cardinalities. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 2 Nov 2006 10:50 David Marcus wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > I'm > > > guessing that by "whole number" you mean natural number, but I really > > > don't know, since you seem to have your own language for everything and > > > you never give definitions for any of the words that you use. > > > > "Whole number" is Cantor's name for his creation. > > > > My question is : Do you maintain omega > n for all n e N? > > Before I can answer the question, I need to know what you mean by the > words/terms. So, please define "omega", "N", and ">". Also, by > "maintain" do you mean that ZFC proves it? For example, Kunen defines the ordinals and the relation "<" for ordinals. He also defines the "natural numbers" as a certain set of ordinals, and he uses omega to denote this set. With these definitions, ZFC proves that for all n in omega, n < omega. So, if that is what your question means, there is your answer. -- David Marcus
From: Dik T. Winter on 2 Nov 2006 10:54
In article <1162470537.261044.229200(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > In article <1162405329.073198.286680(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > The problem is that: > > > For the diagonal number of Cantor's list it is not sufficient to come > > > arbitrarily close to a number which is different from any list number > > > --- or avoid to do so. . > > > > You lost me here. Numbers do not come arbitrarily close to each other. > > Numbers are fixed entities. Sequences can come arbitrarily close to > > each other. > > Irrational numbers are sequences. No. They are equivalence classes. And the representatives are sequences. > > And, representatives are > > *not* limits. When considering the equivalence classes, most sequences > > have one limit: the equivalence class it is sitting in. > > > > I think that you are still thinking that *some* represenation defines > > a real number; but that is not the case. > > In Cantor's list there are those unique representations required. Yes, so what? > > That is especially not the > > case when you consider only representations to some integral base. > > There are other methods to define numbers. You do not like to call > > them numbers, but ideas. But you can not prevent me to call something > > like sqrt(2) a (real) number. And in common mathematics that is just > > what it is. > > > > By definition, every sequence (use any definition, I know Cantor, Sorry, I said definition here, that must have been theorem. > > Dedekind, Baudet and Weierstrass, they all lead to the same): > > {sum{k = 1...n} a_k/10^k} > > is a representative of a "number". It is just a sequence of rationals. > > Therefore I do not understand why you say "Numbers are fixed entities". > They are merely defined by sequences. My error. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |