Cantor Confusion
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From: William Hughes on 9 Oct 2006 09:00 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > Ok. Let's call a day a "numbered day", if we are able > > > > to associate the day with a specific natural number. So day > > > > 5,341,134,322, is a numbered day, but the present day is not > > > > a numbered day. The question is now: "Can X write about > > > > each numbered day?" > > > > > > The question is easy to answer, but this X is a poor example, because > > > there are far better ones like Tristram shandy and the vase, yielding > > > sharper contradictions. > > > > > > 1) Every ball will have left the vase at noon. > > > 2) At noon there are more balls in the vase than at any time before. > > > > > Note, the question originally asked was very careful to > > distinguish between the questions " Will the whole autobiography > > be written?", and "Will certain pages of the autobiography > > be written?, so my repharasing is accurate. > > Yes, but the assertion of Fraenkel and Levy was: "but if he lived > forever then no part of his biography would remain unwritten". That is > wrong, because the major part remains unwritten. You see it by havin > Tristram Shandy write only his firsts of January at unchanged speed. > > > > In terms of the ball problem the question becomes: > > "For each numbered ball is there a time before noon at > > which the ball will be removed?" Answering this question > > "Yes", does not lead to contradictions, > > contradictions (even in your terms) > > only occur if we talk about what happens at noon. > > However, we have accepted potential infinity (i.e. it > > makes sense to talk about events that are certain to take > > a finite amount of time, even if this finite amount of time > > is arbitrarially large). > > With potential infinity there is no contradiction. There it is > meaningless to consider noon, i.e. to consider the completed set, i.e. > to consider every ball. > > If, however, the whole set of N is considered as actually existing, > then there is a contradiction, because then the union of all natural > numbers is a fixed set which does not leave room for further numbers. > Then "each" is contradictive because we know that there is a set of > numbers which is not removed and which has a larger (precisely: not a > smaller) cardinal number than the set of numbers removed. > > > > > In my view we have not gotten very far. We still have > > the result that there is no list of all real numbers > > That is not astonishing, because there are only those few real numbers > which can be constructed. Few? Few compared to what. The real numbers that cannot be constructed? According to you they don't exist. But even these "few" real numbers cannot be listed! > > > (we need to reinterpret our terms, real numbers are > > computable real numbers, and a list is a computable > > function from the natural numbers to the (computable) real > > numbers). > > > > If it gives you a warm fuzzy to say that > > "Every ball will be removed at some time before noon", > > No. To say that every ball will be removed, is wrong, because there is > not every ball. > If it gives you a warm fuzzy to say "For any natural N, the ball numbered N will be removed from the vase before noon" but not to have to say "Every ball will be removed at some time before noon" knock youself out. - William Hughes
From: Han de Bruijn on 9 Oct 2006 09:00 Tonico wrote: > Han de Bruijn wrote: > Mainstream mathematics doesn't slap the hand by which it's fed. And > Set > >>Theory is the _idealization_ of Karl Marx's "ungeheure Waren-sammlung". >>The latter is most characteristic for our society. Thus the undermining >>of Set Theory has its mirror in undermining our society as such. Which >>explains the exceptionally strong emotions accompanying these debates. > > ********************************** > Hahahaha....!! Hohohohoho...!! And I bet Han is serious with this > delicious """"theory"""" of his! Hahahaha...oh, dude. Thanx the hollie > mollie I read the whole message this time....what a laugh!!! And I bet > HdB must have a face as serious as one pound of garlic when he spews > this words...hahahaha! > Well, after this peak of absurdity and ridiculous nonsense reacher by > Han I think this thread can go and lay down in peace: I shall not be > participating oin it anymore, though here and there perhaps I shall > read some message....man, my stomach still hurts!! > Happily laughing still > Tonio > Ps If you don't mind I shall be treasuring in my archives this little > pearl of internet guaka-guaka to show my students and peers....:) Oh, I know very well this little piece of Marxism sounds unbelievable. Han de Bruijn
From: mueckenh on 9 Oct 2006 09:02 Alan Morgan schrieb: > In article <1160296113.211935.299880(a)c28g2000cwb.googlegroups.com>, > <mueckenh(a)rz.fh-augsburg.de> wrote: > > > >Virgil schrieb: > > > >> > We know, not by > >> > intuition, but by logic, that the vase at any time contains more balls > >> > than have escaped. > >> > >> Absolutely false. > > > >Bold words, but unfortunately simply wrong. The contents of the vase > >increases any time by 9 balls. > > > >> Before any balls are put in the vase, it is empty, and > >> after al balls have been removed from the vase it is equally empty. > >> It is only between these time that there are any balls in the vase at > >> all. > >> > >We have a contradiction n ZFC > >1) Every ball will have left the vase at noon. > >2) At noon there are more balls in the vase than at any time before. > > That would be a contradiction. Fortunatly, you can not conclude #2. > You *can* conclude that at every point before noon there are more > balls in the vase than at any time before, but that's not the same > thing. Before noon != noon. If the set N would actually exist, then (2) would be correct. Before noon the contents grows but at noon it should have dropped to zero? That is a ridiculous requirement, in particular because the same numbers of balls are involved if the balls are enumerated slightly different: 1, 11, 21, ... out instead of 1, 2, 3, .... No. The only possible result is the non-existence of the set N and of tranfinite set theory. > > Consider the even simpler case of putting balls in the vase and > never removing them. Put in one ball at one minute before noon, > another ball at 1/2 minute to noon, another at 1/4 of a minute > to noon, etc. At every point in time before noon there are a > finite number of balls in the vase, but just after noon...... According to your proposal it would be empty. Regards, WM
From: William Hughes on 9 Oct 2006 09:53 Han de Bruijn wrote: > Tonico wrote: > > > Han de Bruijn wrote: > > Mainstream mathematics doesn't slap the hand by which it's fed. And > > Set > > > >>Theory is the _idealization_ of Karl Marx's "ungeheure Waren-sammlung". > >>The latter is most characteristic for our society. Thus the undermining > >>of Set Theory has its mirror in undermining our society as such. Which > >>explains the exceptionally strong emotions accompanying these debates. > > > > ********************************** > > Hahahaha....!! Hohohohoho...!! And I bet Han is serious with this > > delicious """"theory"""" of his! Hahahaha...oh, dude. Thanx the hollie > > mollie I read the whole message this time....what a laugh!!! And I bet > > HdB must have a face as serious as one pound of garlic when he spews > > this words...hahahaha! > > Well, after this peak of absurdity and ridiculous nonsense reacher by > > Han I think this thread can go and lay down in peace: I shall not be > > participating oin it anymore, though here and there perhaps I shall > > read some message....man, my stomach still hurts!! > > Happily laughing still > > Tonio > > Ps If you don't mind I shall be treasuring in my archives this little > > pearl of internet guaka-guaka to show my students and peers....:) > > Oh, I know very well this little piece of Marxism sounds unbelievable. > What? You think that something can sound unbelievable but still be true? If something sounds unbelievable, how do we tell if it is true or not? - William Hughes
From: William Hughes on 9 Oct 2006 09:59
mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > David Marcus schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Hi, Dik, > > > > > > > > > > I would like to publish our result to the mathematicians of this group > > > > > in order to show what they really are believing if they believe in set > > > > > theory. > > > > > > > > > > There is an infinite sequence S of units, denoted by S = III... > > > > > > > > > > This sequence is covered up to any position n (included) by the finite > > > > > sequences > > > > > I > > > > > II > > > > > III > > > > > ... > > > > > > > > What do you mean by "cover"? > > > > > > A covers B if A has at least as many bars as B. A and B are unary > > > representations of numbers. > > > > > > Example: A = III covers I and II and III but not IIII. > > > > > > > But it is impossible to cover every position of S. > > > > > > > So: S is covered up to every position, but it is not possible to cover > > > > > every position. > > > > So, your conclusion is that no finite sequence of I's will cover S. > > Correct? > > > > Is this your entire theorem or is there more to the conclusion? > > My conclusion is: > Either > (S is covered up to every position <==> S is completely covered by at > least one element of the infinite set of finite unary numbers Straight quatifier dyslexia. The fact that "for every x there exists a y such that" does not imply "there exists a y such that for every x" - William Hughes |