Cantor Confusion
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From: Dik T. Winter on 20 Nov 2006 22:30 In article <MPG.1fc5e10844be8cf2989911(a)news.rcn.com> David Marcus <DavidMarcus(a)alumdotmit.edu> writes: > Dik T. Winter wrote: > > In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com> "William Hughes" <wpihughes(a)hotmail.com> writes: > > > So lets count the set of all natural numbers {1,2,3,...} > > > There are no natural number left. So we stop > > > using natural numbers and use ordinals > > > (and to nobody's surprise a few things change). > > > > This is wrong. There is no ordinal needed to count the elements of the > > set of all natural numbers. You can count until you weigh an ounce (;-)) > > but you will never finish. Neither the elements you wish to count will > > be exhausted nor the numbers with which you count. I think this is > > potential infinity. On the other hand, when you ask "how many" elements > > there are in N, you need an infinity (and this is, I think, actual > > infinity). But all this hinges quite closely on the semantics of the > > word "count". If seen as a process, you do not need an infinity; when > > seen as the result of a process, you do need an infinity. In many > > languages (German and Dutch amongst others) there are different words > > for the two meanings, but the meanings are conflated in English. > > Are you discussing languages or math? What would a mathematical meaning > for "count the set of all natural numbers" be? I am discussing both language and maths. In German (as in Dutch) there is a clear distinction between "Abz?hlen" and "Z?hlen". And as you are in discussion with somebody from German origin, it is important to keep this in mind. The first word means the process of counting, the second means counting to get a result. In English for both the verb "to count" is used (and is given as translation in dictionaries). I think "to enumerate" is a better translation of "Abz?hlen". So let met rephrase the position of the opponents: To enumerate the natural numbers you do not need ordinal numbers, this is potential infinity. To count the natural numbers you do need ordinal numbers. This is actual infinity. -- 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 20 Nov 2006 22:34 In article <1163666487.368036.271090(a)m7g2000cwm.googlegroups.com> imaginatorium(a)despammed.com writes: > Dik T. Winter wrote: .... > > > That's the question. By means of axioms you can produce conditional > > > truth at most. I am interested in absolute truth. Axioms will not help > > > us to find it. I don't think we need any axioms. > > > > If you want to find absolute truth you should not look at mathematics. > > Really? There are two groups of order 4; could any truth be more > absolute than that? What is the absolute truth of the axioms and definitions you are using? Within some set of axioms and definitions it is indeed an absolute true. Read what Wolfgang Mueckenheim wrote. "Axioms will not help us to find it." -- 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 20 Nov 2006 22:38 In article <1163670499.235193.87050(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > If you want to find absolute truth you should not look at mathematics. > > Not at that what today is called mathematics, I agree. > > I + I = II is very fine and reliable mathematics. Absolutely true. And > this approach can be put forward --- very far. What do you mean with those symbols? How do you define "+" and "="? What is the meaning of "I" and "II"? -- 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 20 Nov 2006 22:45 In article <1163672167.744087.89590(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > This matrix has length omega and width omega. And its diagonal has > > > length omega. No line has length omega. Therefore the width is larger > > > than any line. And the diagonal is longer than any line. This is > > > impossible. > > > > No, that is very possible. > > If you assert that there is no line longer than the diagonal, you have > good reasons, which can be proved. Right. > If you assert that the diagonal can be longer than any line, then you > have no reasons, because the diagonal consists of line elements and > cannot be where no line is. So your second assertion is outside of > logic and outside of any mathematics. Therefore I am not willing to > discuss this topic further. You do not want to discuss it because you are not able to prove it. But indeed, let's stop this non-discussion. Unless you come with a proof. You are not even able to comprehend potential infinity. > The correct result: There must be at least one line which is exactly as > long as the diagonal. There must be an infinite natural number. Never ever do you give a proof. Only assertions. -- 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 20 Nov 2006 22:57
In article <1163672612.950702.12250(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1163604980.169629.197680(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > But you do not allow elliptic or hyperbolic > > > > geometry? If not, why not? > > > > > > I do not forbid it. It is clear that already the simple geometry on a > > > sphere does not yield two parallels. Euclid simply did not consider > > > such kind of plane. In another system we have other axioms (or better > > > fundamental truths). > > > > Well, the same in set theory. In one form it has AC as a fundamental > > truth, in another version it is not a fundamental truth. What is the > > essential difference between the cases? > > > That there is not an "underlying plane" which it is related to. So there should be an "underlying plane", whatever that may mean? What is the "underlying plane" of hyperbolic geometry? What is it in elliptic geometry? > > > > I do not ask what you think. Reread my question. What is true > > > > about a set of numbers in nature or reality? > > > > > > There are so many truths. Take order, 1 < 11, commutativity of addition > > > and multiplication, n + m = m + n. These things do not become invalid > > > or to be proved only because matrix multiplication or quaternions were > > > invented. > > > > Again, no answer. > > If you cannot understand this answer, then we should stop here. No I do not understand it. It does not explain at all what is true about a set of numbers in nature or reality. So you can stop. > > > I think there is a great difference. It is not necessary to call > > > negative solutions "false" solutions as even Descartes did, (because it > > > was customary at his time. Although this custom was justified as long > > > as only positive numbers were called numbers.) But it is necessary to > > > distinguish between negative and positive numbers or real and complex > > > numbers or Euclidean and non Euclidean spaces. > > > > And you were vehemently objecting at calling the irrational numbers > > numbers. > > The reason is that these "numbers" have no decimal representation, > hence cannot be used to prove that they are uncountably many. So what was your vehemence directed at? Not at calling them numbers, because you do so yourself. That they have no decimal representation? But that was not in discussion at that time. So I wonder. > > What is the essential difference between all those cases? I once asked > > you for a definition of "number", and you did never supply a proper > > definition. Now you say yourself that the definition has been changed in > > the course of time. I may note that there is no trichonomy between the > > complex numbers, it is not possible to consistently order them. > > You may call these entities numbers or not. My opposition stems from > their use in Cantor's list and the lacking trichotomy. So, now we may call them numbers (as you do). And, we are back to basics. Your misunderstanding of Cantor's argument and whatever. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |