Cantor Confusion
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From: mueckenh on 19 Oct 2006 05:38 Virgil schrieb: > In article <1161182168.538010.12650(a)f16g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > N = omega = aleph 0. How should one of them have different character? > > According to Cantor, set theory deals with actual infinity, not with > > potential. > > Cantor died in 1918. Things have progressed a bit since then. Not the knowledge about infinity. On the contrary, it has been forgotten and has been applied carelessly and false. Regards, WM
From: Mike Kelly on 19 Oct 2006 06:25 Han de Bruijn wrote: > Dik T. Winter wrote: > > > I think that, compared to Cantor, in modern set theory potential and actual > > infinity are split up again. The contents of the set N form only a potential > > infinity, on the other hand, the *size* is an actual infinity. > > And you call _that_ "thinking" ?! > > Han de Bruijn Hey look, Han is resorting to insults. What a hypocrite! -- mike.
From: William Hughes on 19 Oct 2006 07:05 mueckenh(a)rz.fh-augsburg.de wrote: > jpalecek(a)web.de schrieb: > > > > > > > Every list of real numbers supplies a diagonal number which > > > > is not contained in the list. > > > > > > > > so any way you cut it there is no complete list of real numbers. > > > > > > And there is no complete list of computable numbers. But they are > > > countable. Hence the diagonal argument does not prove anything. > > > > This is nonsense. You say "my proof is flawed, therefore your proof is > > flawed". You'd better think more about your proof. > > In order to avoid computable lists and to have a common base think of > constructible numbers. A constructible number is a number of which > every digit can be obtained by a given formula. Every diagonal number > of a given list is a constructible number. > > The set of constructible numbers is obviously countable. No. There is no list of constructible numbers (your definition). Therefore (by definition of countable) the constructible numbers are not countable. - William Hughes
From: Dik T. Winter on 19 Oct 2006 10:42 In article <eh6oh3$qov$1(a)mailhub227.itcs.purdue.edu> Dave Seaman <dseaman(a)no.such.host> writes: > On Thu, 19 Oct 2006 00:49:13 GMT, Dik T. Winter wrote: > > In article <eh5et9$4gv$1(a)mailhub227.itcs.purdue.edu> Dave Seaman <dseaman(a)no.such.host> writes: > > > On Wed, 18 Oct 2006 14:14:33 GMT, Dik T. Winter wrote: > > > > In article <J7B4p3.ItG(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > > ... > > > > Gesammelte Abhandlungen, Hildesheim, 1962, p. 213: > > > > ... der Unterschied ist nur der, da?, w?hrend die Mengen erster > > > > M?chtigkeit nur durch (mit Hilfe von) Zahlen der zweiten > > > > Zahlenklasse abgez?hlt werden k?nnen, die Abz?hlung bei Mengen > > > > zweiter M?chtigkeit nur durch Zahlen der dritten Zahlenklasse, > > > > bei Mengen dritter M?chtigkeit nur durch Zahlen der vierten > > > > Zahlenklasse u. s. w. erfolgen kann. > > > > or translated: > > > > ... the difference is only that, while sets of the first > > > > cardinality can be counted only through (with the aid off) > > > > numbers of the second class, the counting of sets of the second > > > > cardinality only through numbers of the third class, with sets > > > > of the third cardinality only through numbers of the fourth > > > > class, etc. > > > > > From: > > > > ?ber unendlichen lineare Punktmannigfaltigkeiten, Nr. 6, sec. 15. > > > > I may be wrong, but I interpret the quoted passage to mean: > > > It depends on the exact meaning of "abgez?hlt werden k?nnen". At least > > in Dutch such a sentence means assigning a number to the first, assigning > > a number to the second, etc. I would be pretty surprised if it does mean > > something else or when Cantor means something else with it. But I will > > look tomorrow whether I can find a clarification in his papers. I have checked in the dictionary, and indeed, just like in Dutch there are also in German two verbs with the meaning 'count': z?hlen and abz?hlen. The first in general meaning obtaining the total number of objects you are counting, while the second is more or less the *process* of counting (one of the possible meanings is 'to count down' at the launching of a rocket). So also when you have obtained the number of objects you are likely to use 'gez?hlt' and not 'abgez?hlt'. > I read that to mean that the correspondence goes like this: > > 0 -> w > 1 -> w+1 > 2 -> w+2 I think not. My interpretation is that if you have an (ordered) set of cardinality aleph-0 that you can assign numbers to the elements using the ordinals of the second class. (I may note that here Cantor assumes that all sets can be well-ordered.) > > and so on, where all ordinals of the second number class appear in the > second column. And that is clearly not the case. It is talking about the counting of sets with cardinality aleph-0, so sets with ordinality w, w+1, etc., and when you *do* count them not all ordinals of the second number class will appear. > > But this would precisely explain the problems Wolfgang Mueckenheim has > > with this at all. > I haven't been following that closely, but Cantor's language here is > vague in the sense that some quantifiers need to be inserted to make the > meaning precise, and there is more than one way to do that. I am guided > mainly by a sense that Cantor would not make a Mueckenheim-like mistake > in explaining cardinalities. I see no other possible interpretation in the original German (at least, when German works approximately the same as Dutch). > > > I may be wrong, but I interpret the quoted passage to mean: > > > 1. The cardinality of the set of all finite ordinals is aleph_0. > > > 2. The cardinality of the set of all ordinals having cardinality > > > aleph_0 is aleph_1. > > > 3. The cardinality of the set of all ordinals having cardinality > > > aleph_1 is aleph_2. > > > and so on. This fits with the quotation I provided yesterday. Perhaps, but that would be a strange use of the word 'abgez?hlt'. > > Yes, that was from a title in the Dover edition, section 16. I do not > > know how it relates to his original works. His original works were in > > Mathematischen Annalen in a sequence of articles. That is were I quote > > from. But I see the Dover edition is also in our library, so I will > > also check that. The Dover edition covers pages 282 to 351 of the 'Gesammelte Abhandlungen', originally published in Math. Annalen 1895 and 1897. The quote I gave comes from earlier publications, 1879-1884 (pages 139 to 247 of the Abhandlungen). In that time he may have changed his position quite a bit (note that also during that time he found the diagonal proof). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Randy Poe on 19 Oct 2006 11:07
mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > > > The cardinal numbers of the sets of balls residing in the vase > > > > Crucial phrase missing: The cardinality f(t) of the set of balls in > > the vase at time t STRICTLY LESS THAN ZERO... > > Either we can conclude from f(t) on f(0) > Or we canot at all talk about infinity. That's an incorrect statement. The fact that we have a function such that f(0) is not equal to lim(t->0-) f(t) has nothing to do with whether we can or can't talk about infinity. Discontinuous functions such as step functions are perfectly common in mathematics, physics and engineering. > > > are also > > > natural numbers. f(t) = 9, 18, 27, ... which grow without end. How can > > > such a function take on the value zero? > > > > Because the set of balls at t=0 is not one of these sets. t=0 is not > > a time strictly less than zero. > > I agree. But then the set at t = 0 is undefined. No, the fact that an object is not a member of a particular set does not mean the object is undefined. If I tell you my pet is not a member of the set of dogs, that does not mean I know nothing about my pet or that my pet is undefined. - Randy |