Cantor Confusion
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From: mueckenh on 8 Oct 2006 08:01 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > You got it several times already. According to the axiom of infinity > > the set of all natural numbers does exist. And with it the following > > statements are true: > > > > 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". Regards, WM
From: William Hughes on 8 Oct 2006 10:36 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. 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). In my view we have not gotten very far. We still have the result that there is no list of all real numbers (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", but to not have to say that "the vase is empty at noon", knock yourself out. - William Hughes
From: Albrecht on 8 Oct 2006 10:53 Virgil wrote: > In article <1160223586.604282.269450(a)h48g2000cwc.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > Tonico wrotef: > > > > > Maths, just like the other sciences, isn't grounded on dogma, and > > > people forwarding REASONABLE, well-based objections, opinions or ideas > > > on whatever are always welcome. > > > > _There_ is your problem! Ask Virgil, and the other mathematicians here: > > > > MATHEMATICS IS _NOT_ A SCIENCE > > > > And therefore there is NO guarantee that it "isn't grounded on dogma". > > > HdB's version of mathematics certainly seems to be grounded in dogma. > > My mathematics is merely grounded on determining what can be derived > from a given set of axioms. 1. The moon exists 2. The moon consits of green cheese 3. The man in the moon is a mouse Now do maths. Else explain why you prefer a special set of axioms. Best regards Albrecht S. Storz
From: David Marcus on 8 Oct 2006 12:11 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 or that you think mathematics should be done differently? > Or > S is not covered up to every position by unary naturals ==> The > positions of S are not defined ==> S does not exist. > > There is no actual infinity, but nly potential infinity, i.e., S is not > complee but only has as many bars as you or anothe one can count. -- David Marcus
From: David Marcus on 8 Oct 2006 12:30
mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > You got it several times already. According to the axiom of infinity > > > the set of all natural numbers does exist. And with it the following > > > statements are true: > > > > > > 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. Mathematics doesn't include "balls", "vases", "enter", "escape". Mathematics uses sets, functions, numbers. Actually, please state the problem in words, then state how you translate it into (or model it using) mathematics. -- David Marcus |