Cantor Confusion
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From: Jesse F. Hughes on 9 Oct 2006 07:13 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > 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. What an absolutely adorable theory! -- "Your knowledge is the power that promote good thought, how then can you have good thought without powerful knowledge or how can you have powerful knowledge without learning or how can you learn without a teacher and how can a teacher teach if he or she has not learned the subject." --CA Alternative High School
From: Tonico on 9 Oct 2006 08:10 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. > > Han de Bruijn ********************************** 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....:)
From: mueckenh on 9 Oct 2006 08:45 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. > (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. Regards, WM
From: mueckenh on 9 Oct 2006 08:51 David Marcus schrieb: > > > > 1) Before noon every ball comes out of the vase. At noon the vase is > > > > empty. > > > > 2) Before and at noon there are more balls in the vase than have come > > > > out. > > > > > > How do you translate the words of the problem into mathematics? > > > > 0) There is a bijection between the set of balls entering the vase and > > |N. > > 1) There is a bijection between the set of escaped balls and |N. > > 2) There is a bijection between (the cardinal numbers of the sets of > > balls remaining in the vase after an escape)/9 and |N. > > > > Instead of "balls", use "elements of X where X is a variable". > > Sorry, perhaps I wasn't clear. That's not what I meant. Please state the > problem using mathematics. I am sure you are able to translate brief notions like "to enter, to escape" etc. by yourself into terms of increasing or decreasing values of variables of sets, if this seems necessary to you. Here, without being in possession of suitable symbols, it would become a bit tedious. Regards, WM
From: mueckenh on 9 Oct 2006 08:54
David Marcus schrieb: > 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 <==> S is > > an unary natural) ==> Contradiction, because S can be shown to be not a > > unary natural. > > Are you saying that standard mathematics contains a contradiction Yes, obviously. > or > that you think mathematics should be done differently? Not mathematics but set theory. Regards, WM |