Cantor Confusion
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From: Sebastian Holzmann on 19 Oct 2006 12:35 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > And MB later again: The balls in a vase problem is not within set > theory. > > You are wrong. If the vase is not empty at noon, than one can never What is the definiton of a "vase" in the language of set theory? What is a "ball"? All that has been done in this thread is a translation of a seemingly real problem into the language of mathematics. If the results of different translations differ, this does not have to be the fault of mathematics. It can be due to the different style of translation. Do not try to eat the cookbook!
From: mueckenh on 19 Oct 2006 12:37 Virgil schrieb: > In article <1161200729.156817.65510(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1161079685.233073.120000(a)k70g2000cwa.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > MoeBlee schrieb: > > > > > > > > There is no "equal weight" in the proof. > > > > > > > > > > > > > You haven't yet noticed it? Each digit of the infinitely many digits of > > > > the diagonal number has the same weight or importance for the proof. > > > > > > It is only necessary that each digit have non-zero weight in order for > > > the difference between the diagonal and one of the list to be non-zero. > > > > > > > > > > In > > > > mathematics, the weight of the digits of reals is 10^(-n). > > > At least in decimal notation, but as that makes all the weights > > > > > non-zero, that is sufficient to distinguish the diagonal from each of > > > the listed numbers. > > > > Only for a finite diagonal. In the infinite case we have for example 1 > > = 0.999... > > But if the diagonal is not allowed to contain any 1's, 0's or 9's, as in > standard constructions of "diagonals", such cases cannot occur. And why can "such cases" occur if the diagonal is allowed to contain these digits? What do you think? > Its digits do not have equal weight, as "Mueckenh" well knows, but each > digit has a non-zero weight, lim {n--> oo} 10^(-n) =/= 0 ??? No, I did not yet know, but your artistic vase calculations may convice me on the long term. Regards, WM
From: Dave Seaman on 19 Oct 2006 12:40 On Thu, 19 Oct 2006 14:42:32 GMT, Dik T. Winter wrote: > 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 won't dispute the meaning of the verb, but it seems very strange to me that Cantor could make such a fundamental mistake. Am I right that "sets of first cardinality" are the finite sets, "sets of second cardinality" are the denumerable sets, and so on? I have not found an explanation of that terminology, although "numbers of the second class" are clearly explained. > > 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. I don't follow you here. I was the one describing the table, and according to my intended description, it is definitely the case that every number of the second number class appears in the right-hand column. Perhaps I did not make myself clear enough. Here is a slightly expanded version of what I intended: 0 -> w 1 -> w+1 2 -> w+2 ... w -> w2 w+1 -> w2+1 ... w2 -> w3 w2+1 -> w3+1 ... w2+w -> w3+w ... and lots more. I have to stop somewhere, but my intent is that the right-hand column does indeed contain every number of the second number class, and that the entire table therefore contains aleph_1 rows. What German word would you use to describe the contents of the above table? Notice that the table does not have an end. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: mueckenh on 19 Oct 2006 12:45 Dik T. Winter schrieb: > Again replying to more than one article at once without giving proper > references. That makes it very difficult to follow threads. I am sorry, I have only 15 shots per several hours, but far more opponents. I repeat only that part of the discussion which I answer to. > Yes, according to Cantor. And according to modern set theory that is wrong. > To count the elements of a set with the ordinal omega you need only the > finite ordinals. I do not understand you. We cannot count all naturals without saying "omega" and being finished. > 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. But in modern set theory omega = {1,2,3,...} is the contents and aleph_0 = omega is the size of the set. Modern set theory has no use for potential infinity. > This one is in reply to <J7BuCw.LJr(a)cwi.nl>: > > In article <1161201985.224100.146920(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > Let me ask. You *did* state that "The function f(t) = 9t is continuous, > > > because the function 1/9t is continuous", yes? Do you not see that that > > > reasoning is wrong? That if 1/f(t) is continuous that does not mean that > > > f(t) itself is continuous? > > You can not answer this question? See below. > > > > > > Moreover, when we let t go to infinity, 1/9t is *not* continuous at > > > > > infinity (whatever that may mean). We can only define the limit, > > > > > not the function value. > > > > > > > > We can only define limits in *all* cases concerning the infinite. > > > > Nothing else is possible. > > > > > > Yes, so you can not talk about continuity. > > > > We can talk about continuity in the finite domain. We can calculate the > > limit in case of f(t) --> 0 and we can from that define a limit oo for > > the case 1/f(t). > > I have my doubts about the latter, because oo is not a real number, and > doing arithmetic with that one is impossible. > > > From these procedures we can obtain that the function > > 1/f(t) will never yield the limit 0. That is the continuity requirement > > I mentioned. > > You are doing arithmetic with oo as if it is a real number. "As if" it is larger than any real number. But, yes, according to LAUGWITZ we can use it as a very large number. In any case I am sue that oo > 1 is valid and can be used. > However, in > standard mathematics you can state that lim{t -> oo} 1/f(t) = 0. That does > *not* mean that 1/f(oo) equals 0, because that one is not defined, neither > does it state that there is continuity, and also it does state nothing about > the value of f(t) at t = oo, because that is also not defined. Then we cannot say anything about f(oo). That is in fact the truth, but not in set theory. Set theory says about oo: omega is finished. > > ===== > This one is in reply to <J7BuGM.Lup(a)cwi.nl>: > > > > > Any constructable list of constructable > > > numbers gives a constructable diagonal number, > > > so any *constructable* > > < list of constructable is necessarily incomplete. > > > > Any list gives a constructrible diagonal number. Any diagonal number is > > constructed: > > Yes. You can construct a number from any list, but you can not construct > a constructable number from a list that is itself not constructable. Every diagonal number is constructed and, therefore, is constructible. > For a constructible number the whole process to derive that number should > be construction. The list is the first step in derriving that number, so > for the diagonal to be constructable, that list must be constructable. No. > > > Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976), p. > > 54: "Why, then, the restriction to the digits 1 and 2 in our proof? > > Just to kill the prejudice, found in some treatments of the proof, as > > if the method were purely existential, i.e. as if the proof, while > > showing that there exist decimals belonging to C but not to C0, did not > > allow to construct such decimals." > > What are C and C0 in this context? p. 52 explains it: "Lemma. Given any denumerable subset C0 of C, there exist members of C which are not contained in C0; that is to say, C0 is a proper subset of C." So C is the continuum R and C0 is the set of list numbers > > > If you cannot understand that, then try this: The set of diagonal > > numbers is countable. Therefore the diagonal proof proves the > > uncountability of a countable set. > > Why is the set of diagonal numbers countable? How do you derive that? Because only a countable set of numbers can be constructed using a language with a finite alphabet. Regards, WM
From: William Hughes on 19 Oct 2006 12:48
mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > A constructible number is a number which can be constructed. Definition > > > obtained from Fraenkel, Abraham A., Levy, Azriel: "Abstract Set > > > Theory" (1976), p. 54: "Why, then, the restriction to the digits 1 and > > > 2 in our proof? Just to kill the prejudice, found in some treatments of > > > the proof, as if the method were purely existential, i.e. as if the > > > proof, while showing that there exist decimals belonging to C but not > > > to C0, did not allow to construct such decimals." > > > > > > Definition (by me): A number which can be constructed like pi, sqrt(2) > > > or the diagonal of a list is that what I call constructible. If you > > > dislike that name, you may call these numbers oomflyties. Anyhow that > > > set is countable. > > > > Nope. By the definitions you use, that set is not countable. > > > Every set of constructions is countable due to the finite alphabet of > any language. No. If you restrict yourself to computable functions you have some counterintuitive results. Assume that the language you are working in has a finite alphabet. Then the set of all finite strings in the language is listable using a computable function (use dictionary order). And so the set of all finite strings is countable. Now, A, the set of all strings which define a computable number is a subset of the set of all finite strings. So A is countable, right? Wrong! It is not true that every subset of a countable subset is countable. It is not true that there is a computable function which will list the elements of A (to do this you would have to be able to identify the elements of A, and to do this you have to solve the halting problem). > > > > And that set cannt be listed. > > > > And here is your problem. Uncountable means unlistable. > > Not my problem. Countably infinite means unlistable too. Yes, but what does unlistable mean? - William Hughes |