Cantor Confusion
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From: Dik T. Winter on 18 Oct 2006 19:06 In article <1161182168.538010.12650(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > One of the most serious errors can he found in the statement that > > "to count sets of first cardinality you need ordinals of the second > > class" > > where (I may have first and second wrong), first cardinality means (in > > current terminology) aleph-0 and second class ordinals means (in current > > terminology) omega and larger (until omega^omega or somesuch). The > > error is of course that there is a set of cardinality aleph-0 that can > > be counted using finite ordinals only (the natural numbers). > > Numbers of first numberclass are the finite (natural) numbers (0), 1, > 2, 3, .... To count them, you need a number of the second class, > aleph_0 or omega. 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 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. -- 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 18 Oct 2006 19:19 Ah, you are responding to many articles at once without providing proper references. From the first article <J79F9o.AnA(a)cwi.nl>: In article <1161182462.511657.72280(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > It is impossible to show a "valid" mathematical proof against set > > > theory. > > > > Ah, so you agree that you can not prove an inconsistency using mathematical > > terms of proof. > > I agree that set theorists will ever accept any proof of inconsistency > as valid. > > > > > We have discussed the vase and I would not have believed in > > > advance that anybody could maintain arguments here like Virgil and > > > William and others. > > > > No. You never would believe that anybody would use mathematical proofs > > against your intuition. > > It is not *intuition* to find a proof of lim{n-->oo} n = 0 is wrong. I have not seen such a proof. ===== From the second article <J79Ewq.9M0(a)cwi.nl>: > > In actual mathematics you can state that > > sqrt(2) squared is exactly equal to 2 > > And in actual life you can state that all problems are exactly solved. > I wonder whether our politicans should not get a better mathematical > education. I wonder how you can not see the difference between something theoretical (like actual mathematics) and something else. ===== From the third article <J79FHw.BCu(a)cwi.nl>: > > Redo your calculations. With 100 bits there are 2^100 possibilities, so > > the cardinality for the set of numberss represented is <= 2^100. > > Here is a simplified problem: With one bit here are two numbers > possible, namely 0 and 1, but the set of numbers realized with one bit > has cardinality 1, namely either the number 0 or he number 1 but not > both can be realized. And now you are just obfuscating on purpose. -- 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 18 Oct 2006 19:47 Again replying to more than one article at once without giving proper references. That makes it very difficult to follow threads. 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? > > > > 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. 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. ===== 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. 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. > 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? > 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? -- 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 18 Oct 2006 20:49 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. But this would precisely explain the problems Wolfgang Mueckenheim has with this at all. > 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. 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. -- 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 18 Oct 2006 21:14
In article <1161183237.727249.154740(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > In article <1161079802.120515.175530(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > The inconsistency is that > > > 1) For the balls inserted until noon, you can find the result: It is > > > the set N. > > > 2) For the balls removed until noon, you can find the result: It is the > > > set N. > > > 3) For the balls remaining at noon, the same arguments of continuity > > > which lead to (1) and (2) cannot apply. > > > > There are quite a few obvious reasons. > > (1) 1) is not because of continuity The reason that continuity plays no role is because the function of the number of balls in the vase when written as a function of t is discontinuous at infinitely many positions. > Why then? Depending on the person who writes about the problem. One possibility is defining a set of functions dependent on time t that tells whether at time t (not t now goes from -1 to 0) ball n is in the vase or not. Addition of those functions gives the number of balls present in the vase at time t. But all depends on the exact mathematical formulation of the problem. There is none (in my opinion, but I have already had discussions with others about this point, so I will not repeat them). > > (3) no continuity reasoning can lead to the result that the balls > > remaining at noon is the set N. > > But more than 1. No continuity reasoning can lead to anything. The functions are inherently not continuous. When we look at the functions when we let t go from -1 to 0 we see: Balls added since time t = -1: entier(- log_2(- t)) * 10 Balls removed since time t = -1: entier(- log_2(- t)) Balls remaining since time t = -1: entier(- log_2(- t)) * 9 None of these functions is continuous, and in fact they have discontinuties infinitely often within any time frame that contains t = 0. Moreover, none is defined at t = 0. They all have a singularity at t = 0 (and are not extensible in a sensible way beyond t = 0, unless you allow for imaginary balls, but I have no idea what the entier function does with imaginary values). You can *model* the problem in different ways in mathematics, and one of those models leads to 0 balls in the vase at time 0. And there are presumably other models that give different results. So, unless the problem is stated in a mathematically proper way, actually nothing can be said about it. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |