Cantor Confusion
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From: Virgil on 18 Oct 2006 22:03 In article <1161201626.885258.106970(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Sebastian Holzmann schrieb: > > > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > > Further the result o the gedankenexperiment must not depend on > > > switching numbers. Removing balls 1, 11, 21, ... does not change the > > > quantities in fact, but according to set theory it does. Therefore set > > > theory has been contradicted. > > > > No. In mathematics, there is no such things as "balls" or "vases". You > > have tried to model some vaguely "real" problem in the world of > > mathematics. > > We had this objection already. Translate balls and vase into elements > and set. The result will not change. It yields a contradiction in any > case, i.e., independent of its result. Not in standard mathematics. We can have a function whose domain is some open real interval of time including noon and whose codomain is the power set of N and whose value at t is the set of balls in the vase at time t, but since the codomain is not a topological space, there can be no issues of continuity. Any such function conforming to the reqirements of the problem will produce the empty set at or after noon.
From: Virgil on 18 Oct 2006 22:17 In article <1161202439.521429.220310(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Either we can conclude from f(t) on f(0) > Or we canot at all talk about infinity. False dichotomy. > > > > > are also > > > natural numbers. f(t) = 9, 18, 27, ... which grow without end. How can > > > such a function take on the value zero? By being discontinuous at zero, as it already is at every point at which its value changes. > > > > 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. Only if it fails to be defined at every other point as well. > There is no > calculating and reasoning with infinite sets. "Mueckenh" only displays his own limitations, but can not impose them on anyone else. > Bijections between > infinite sets are undefined. They are defined in my world.
From: Virgil on 18 Oct 2006 22:21 In article <1161202733.289699.71980(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > imaginatorium(a)despammed.com schrieb: > > > > > > > > But the function of balls/numbers removed from the vase is a > > > > > continuously (stepwise) increasing one, containing all natural numbers > > > > > at noon? > > > > > > > > Uh, yes, unless I mysteriously misunderstand you... If takenout() is a > > > > function from time to the power set of the integers (i.e. it maps to a > > > > set of integers) then each natural number m is included in the set that > > > > takenout() maps to from time = -1/m. So by time zero, all natural > > > > numbers are included. > > > > > > > > Was there a question with that? > > > > > > And this result would change, if the numbers of the balls were > > > exchanged, for instance multiplied by 10 after having been inserted? > > > > Can you clarify what "exchanged" means? Obviously, if you do exactly > > the same thing with exactly the same balls, it makes no difference what > > is written on them, > > That is an astonishing statement, but I welcome it and agree. Why do > you speak of exactly the same balls, however? They are all > indistinguishable like atoms. How do you know that? I do not recall anything in the statement of the problem which requires the balls stripped of their numbers be indistinguishable, so that such an assumption is unjustifiable.
From: Dave Seaman on 18 Oct 2006 22:39 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 read that to mean that the correspondence goes like this: 0 -> w 1 -> w+1 2 -> w+2 and so on, where all ordinals of the second number class appear in the second column. Although it's true that the first column consists entirely of numbers of the first two number classes (finite and denumerable ordinals), the table itself has aleph_1 rows, and therefore a number of the third number class is needed to represent the totality of the counting process. > 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 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. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: David Marcus on 18 Oct 2006 22:49
Han de Bruijn wrote: > David Marcus wrote: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>David Marcus schreef: > >>>stephen(a)nomail.com wrote: > >>>>What does it mean for a thing in the real world to be true? > >>>>How do you know if a thing in the real world is true? > >>>> > >>>>Consider the twin slit experiment. Is the fact that none of > >>>>the following accurately describe the situation an inconsistency? > >>>> a) the photon goes through one slit > >>>> b) the photon goes through both slits > >>>> c) the photon goes through neither slit > >>> > >>>In Bohmian Mechanics (and similar theories), the photon goes through > >>>only one slit. Physicists could learn something about logical thinking > >>>from mathematicians. > >> > >>Sure, theories. Can't you talk about something else but "theories"? > > > > In the real world, the photon undoubtedly goes through one slit. > > Not in my real world of physics. > > > However, discussing physics with you is undoubtedly even more useless > > than discussing mathematics. > > BTW. Are you a physicist? Why do you ask? -- David Marcus |