Cantor Confusion
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From: William Hughes on 21 Nov 2006 19:59 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > What about all contructible numbers or all words? They cannot be mapped > > > on N though they are a countable set. > > > > > > Absolute piffle. The question is whether the set N can > > be mapped to the set N, not whether some other > > set can be mapped to the set N. > > It is the same impossibility, but not so obvious. > > > > [Although it is interesting to note that both the constructable numbers > > and the set of all words can be mapped to the set N (please check > > the definiton of countable set)] > > The countability of the set W of all the words of a finite alphabet is > proved by other means than the usual countability proof. The latter > consists in constructing a bijection with N, i.e., in constructing a > list. But it is impossible to construct a list of all words. It is > impossible to construct a bijection between W and N. Why should it be > possible to construct a bijection between N and N? Because a bijection between W and N cannot be the identity map but a bijection between N and N can be the identity map. > > > > > > > Note however that in order to attach a definite cardinal number to > > > every set, well-ordering must be possible for every set. That is why > > > Cantor insisted on well-ordering of all sets. > > > > Piffle. Well-ordering has nothing to do with the existence or not of > > at least one bijection. > > > You are wrong. If a set cannot be well-ordered, it cannot be assigned a > definite cardinal number. In order to assign a cardinal, at least one > bijection to an ordinal is required. This requires at least one > well-ordering. There are two possible defintions for the cardinals. One is the set of equivalence classes sets under the equivalence relation bijection. The second is to restrict these equivalence classes to those classes that contain at least one ordinal. In the second case only sets that can be well ordered have cardinal numbers (Duh!). In any case: any set of natural numbers can be well ordered. So if the set of all natural numbers N exists, it is both an ordinal and can be assigned a definite cardinal. So trichotomy will hold. - William Hughes
From: Dik T. Winter on 21 Nov 2006 23:01 In article <eju5op$l77$1(a)news1.nefonline.de> Ralf Bader <bader(a)nefkom.net> writes: > Dik T. Winter wrote: .... > > 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 "Abzählen". > > As a native speaker of German, I can't confirm this. There is a difference > between "z?hlen" and "abz?hlen", because one can't use these words > interchangeably, and this difference seems to have something to do with > what you wrote, but it is far less clear-cut than your remarks seem to say. I know it is not as clear cut. In Dutch we have the same distinction as in German. At least in Dutch, and also in German I think, "tellen" ("z?hlen") almost always refers to getting the number of articles. So when you "z?hl" the elements of a set you want to know the cardinality of the set. On the other hand, when you "z?hl" the elements of a set "ab" ("aftellen" in Dutch) you are assigning labels to the elements of the set, and I think this is also the case in German. So when it is stated (in German) that a set is "abz?hlbar" it means that you can assign labels (from the set of integral numbers) to each element of the set. On the other hand, when you can assign a number to the size of the set you might state that it is "z?hlbar". And I think that it is this very distinction that makes the difference between potential and actual infinity strong for the German speakers (and the Dutch), but not for the English speakers (because they do not have that distinction with the word count). > However this may be, in discussions with M?ckenheim it doesn't matter > anyway, because M?ckenheim plainly doesn't understand German, or, more > precisely, Cantor's original papers. I know. Whenever I state that the paper that contains the diagonal proof is not about reals, M?ckenheim always comes back with the second part of the first paragraph of that paper to show that it is. > Comparing with Cantor's original, > one notices that Cantor had proved a much more general result, namely that > the real plane stays connected if an arbitrary countable set of points is > left out. The set of points with rational coordinates he later mentions as > an example, and M?ckenheim took that example for the main issue, Pretty similar to my experience. > It simply doesn't make any sense, and doesn't lead anywhere, to discuss > mathematics with M?ckenheim, neither in English nor in German. But it has entertainment value. I still wonder what he will do now that in an older article he wrote that the real numbers were not numbers, but in a later article uses the term "real numbers" to describe them. It also did lead me to read what Cantor actually did write, and I found that he was not without errors in his writing. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Heinz Mau on 22 Nov 2006 01:59 Am Tue, 21 Nov 2006 15:49:57 +0100 schrieb Eckard Blumschein: > When we learn from history we know that Ecki is a little stupid girl who is to dumb to learn anything about mathematics.
From: Heinz Mau on 22 Nov 2006 02:00 Am Tue, 21 Nov 2006 16:23:33 +0100 schrieb Eckard Blumschein: > For the first time I do not feel unnecessarily lectured and deliberately > misunderstood. Oh we are so sorry. Ecki - please go away.
From: Han de Bruijn on 22 Nov 2006 03:22
Eckard Blumschein wrote: > To what extent criticism by WM might be justified? > Let me tell you my position: WM is most likely wrong if he sees the > finished infinities the fundament of modern mathematics. Quite on the contrary. Eckard, please learn more about Wolfgang Mueckenheim in the first place: http://www.fh-augsburg.de/~mueckenh/ Wild guess: his ideas will turn out to be not so uncomfortable with you. Han de Bruijn |