Cantor Confusion
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From: William Hughes on 9 Oct 2006 12:55 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > 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. > > Compared to the assumed set of uncountably many. Funny, you claim that the term "uncountably many" has no meaning, but you use it. > > > The real numbers that cannot > > be constructed? According to you they don't exist. But even > > these "few" real numbers cannot be listed! > > Nevertheless the diagonal proof shows only that there are elements of a > countable set which have not yet been constructed.\ No, it is much stronger. It shows that any list of constructable numbers is not complete. > > > > > > > > > (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" > > There is not "any natural" but only those which we can define. O, so there are now only a finite number of naturals, not even an arbitrarially large number. But you continue to prattle on about limits. > There is > a largest natural which ever will be defined. Hence mathematics in the > universe and in eternity has to do with only a very small sequence of > naturals. > > Writing 1,2,3,... is but cheating > If you want to deal with a system in which there is an unknown but largest natural, knock yourself out. But you have a long way to go before you are even close to being consistent. And don't attempt to use results from this system to say that results from another system are wrong. Note that according to you the ball in vase problem is trivial. At some time, strictly before noon we will reach the largest natural. After this, nothing happens. - William Hughes > Regards, WM
From: Virgil on 9 Oct 2006 12:57 In article <1160397914.738238.238220(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > 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. What part? Tristam has to die, or otherwise be stopped from writing, in order to have any part of his life left unwritten.
From: Virgil on 9 Oct 2006 12:57 In article <1160398309.375726.76250(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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 Try: Let A_n(t) be equal to 0 at all times, t, when the nth ball is out of the vase, 1 at all times, t, when the nth ball is in the vase, and undefined at all times, t, when the nth ball is in transition (times at which a ball changes location). Note that noon is not a time of transition for any ball, though it is a cluster point of such times. let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase at any non-transition time t. B(t) is clearly defined and finite at every non-transition point, as being, essentially, a finite sum at every such non-transition point, and is undefined at each transition point. Further, A_n(noon) = 0 for every n, so B(noon) = 0. Similarly when t > noon, every A_n(t) = 0, so B(t) = 0
From: Virgil on 9 Oct 2006 13:00 In article <1160398472.451589.174540(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Are you saying that standard mathematics contains a contradiction > > Yes, obviously. That it contradicts what "Mueckenh" believes is not evidence that it contradicts itself, which no one has been able to show as yet.
From: Virgil on 9 Oct 2006 13:04
In article <1160398930.398603.113820(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Except that such set theories as ZF and NBG actually exist, disproving "Mueckenh"'s claim that they do not. > > > > 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. Why should a theory that specifically depends on balls being removed apply when none are? > > Regards, WM |