Cantor Confusion
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From: Bob Kolker on 6 Dec 2006 10:41 mueckenh(a)rz.fh-augsburg.de wrote: > > I have understood the concept and its failure. (That is the parallel > between those who have not yet arrived and those who have already left: > Both are not there.) What failure? THe concept of infinite sets has lead to the theory of real and complex numbers, of integration and ultimately to theoretical physics which as made possible the computer on which you spew your nonsense internationally. By their fruits ye shall know them. Mathematics based on infinite sets has produced useful and even indispensible results. Try getting modern physics without it. Bob Kolker
From: mueckenh on 6 Dec 2006 10:53 Franziska Neugebauer schrieb: > Han de Bruijn wrote: > > > Franziska Neugebauer wrote: > >> You still have not yet understood the concept of inifinite sets. > > There may be no person in the world who understands them better than > > Wolfgang Mueckenheim. > > ,----[ http://en.wikipedia.org/wiki/Understand ] > | Understanding is a psychological process related to an abstract or > | physical object, such as, person, situation and message whereby one is > | able to think about it and use concepts to deal adequately with that > | object. > `---- > > He has yet only documented his ability to deal _in_adequately with that > object (infinite sets). This does not support your claim that he does > understand the concept. > > > But I think _you_ still have not yet understood the difference between > > "understanding" and "being reluctant to accept". > > I think that he does not only not understand the concept of infinite > sets but also he refuses to accept this fact. A "fact" of pure tought can be reduced to and understood by neuroscience. But the "facts" of transfinite set theory appear to rest mainly on sociology. Regards, WM
From: Franziska Neugebauer on 6 Dec 2006 10:57 Han de Bruijn wrote: > Franziska Neugebauer wrote: > >> Han de Bruijn wrote: >> >>>But not by people with a mathematics (mis-)education, of course, as >>>I've observed more than once in this newsgroup. Being incapable of >>>common speech and common sense and ... being proud of it. >> >> You know that you are posting to sci.math? > > Yeah, so what? There exist also _well_-educated mathematicians. Which > means that they have absorbed more of the world than just mathematics. > And they do not think that Set Theory is the panacea for all diseases. I never claimed that (some) set theory was the panacea for all diseases. F. N. -- xyz
From: mueckenh on 6 Dec 2006 11:11 William Hughes schrieb: > > > You cannot simply exhange the quantifiers to get > > > > > > There exists a line L, such that for every natural number n, > > > every natural number m<=n, is contained in L. > > > > I can simply do that for any finite linear (= totally orderd) set. > > You have made this comment several times but you have > never said *why* this is true. You need to give some > argument to show why something that is not true in the > general case is true for a totally ordered set. > > Counterexample: > > Consider A= [0,1), the set of real > numbers in greater than or equal to 0 and less than 1. > This set is totally ordered. This set is > composed of finite elements. We have the true statement > > For every element r in A, there exists an element > s in A such that r < s. > > However, if we simple reverse the quantifiers we get > the false statement This set is not finite. > > There exists an element s in A such that for every > element r in A, r<s. > > So the fact that you have a totally ordered set consisting > of finite elements, does not mean that you can reverse > the quantifiers. I said: for any finite linear (= totally orderd) set. This means: for any finte set. Considering natural numbers, we have the coincidence between set of (different) finite numbers and finite set. Recall my simple example: Every set of even natural numbers like {2,4,6,...,2n} has a cardinal number which is less than some number in the set. This theorem does not become invalid if n grows infinitely. A set of even atural numbers cannot have an infinite cardial number. > > > > So I can do it for every line. > > > You need to give some other argument to show that L exists. > > > This you have not done. > > > > > Recall: > > The potentially infinite set of natural numbers > exists. That does not involve the existence of every natural number. (Don't ask me which is missing. Those which can be named can be incorporated.) > > It is possible to have a bijection involving a > potentially infinite set. > > > > > It is in fact easy to show that L cannot exist. > > > > > > If X is such that for every natural number n, n is > > > an element of X, then X is a potentially infinite set. > > > > > > No line L is a potentially infinite set. > > > > > > Therefore, there does not exist a line L such that > > > > > > for every natural number n, every natural number > > > m <= n is an element of L. > > > > Therefore, there is no set of all finite lines, i.e., there is no set > > containing all natural numbers as elements. > > However the potentially infinite set of all finite lines exists. > It is possible to have a bijection involving the potentially > infinite set of all finite lines. > > > > > > > > > > The fact that there each natural > > > number is a member *some* line, does not > > > mean there is one line which contains all > > > natural numbers. > > > > > > The point remains. > > > > > > A set with a largest element can have elements > > > all of which are natural numbers. > > > > > > A potentially infinite set without a largest element > > > can have elements all of which are > > > natural numbers. > > > > Correct, although this set can never be considered complete. > > Irrelevent. It does not matter whether we can > consider a potentially infinite set "complete" or not. All that > matters is that we can have a bijection involving a potentially > infinite set. Not a complete bijection. A bijection is but a set of ordered pairs. If the domain is incomplete (and the range too), then the bijection cannot be complete. > > > > > I claim that every element of the diagonal must be an element of a line. > > The statement > > Every element of the diagonal must be an element of a line. > > is true. But the statement > > Every element of the diagonal must be an element of a single line. > > is false. But it cannot be false for a finite line, because finite sets allow quantifier reversal. Now there is no infinite line. > Therefore we cannot use the statement > > Every element of the diagonal must be an element of a line. > > to conclude that there is a bijection between the diagonal and a single > line. > > Do you intend to keep claiming that a bijection can exist > between the diagonal and a single line?. If the diagonal would actually exist, then a line of same length would actually exist. > > > > > > If the "infinite set of finite numbers" existed, then we would have a > > bijection between the diagonal and a line. > > > > We have agreed to disagree on whether the "infinite set of finite > numbers" > exists. However, we have agreed that the "potentially infinite set > of finite numbers" exists. But we have not agreed that it exists actually (i.e., is complete). > > > > (A diagonal cannot exist without being in a line.) > > False. Every element of the diagonal must be in some > line. However, there is no line that contains all elements of the > diagonal. So the diagonal is not in a line. > The elements of the diagonal form the potentially infinite > set of natural numbers. This potentially infinite set exists, > so the diagonal exists. So the diagonal can exist without > being in a line. Why don't you directly say that the potentially infinite diagonal actually exists? Regards, WM
From: mueckenh on 6 Dec 2006 11:15
Bob Kolker schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > I have understood the concept and its failure. (That is the parallel > > between those who have not yet arrived and those who have already left: > > Both are not there.) > > What failure? One of many examples: The set {2,4,6,...,2n} has a cardinal number less than some numbers in the set. This does not change when n grows (yes, it can grow!) over all upper bounds. Therefore the assertion that the set of all even natural numbers has a cardinal number gretaer than any even number is false. >Te concept of infinite sets has lead to the theory of > real and complex numbers, of integration and ultimately to theoretical > physics which as made possible the computer on which you spew your > nonsense internationally. > > By their fruits ye shall know them. Mathematics based on infinite sets > has produced useful and even indispensible results. Try getting modern > physics without it. For applications the usual way to interpret things is sufficient. Regards, WM |