Cantor Confusion
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From: Virgil on 16 Oct 2006 16:02 In article <1161008178.842467.106720(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > But the end time of the problem (noon) does not correspond to > > an integer (neither in standard mathematics, nor in your > > system, whether or not you interpret the problem as dealing > > with infinite integers as well as finite integers). So the function > > 9n does not have a value at noon. There is no way > > it can be continuous at noon. And since there is no > > value of n that corresponds to noon, 9n cannot be used > > to determine the number of balls in the vase at noon. > > But the function n can be used to determine the number of balls removed > from the vase at noon? No. Since your n has not been related to time, your n can tell us nothing about what happens at what times. > > Regards, WM
From: Han.deBruijn on 16 Oct 2006 16:02 David Marcus schreef: > Han de Bruijn wrote: > > > > Sigh! Start digging into my website. I've said more about mathematics > > than anybody else in 'sci.math'. > > I'm sorry, but I looked at your website and couldn't find anything about > Mathematics. If you do have something to say, please say it here and say > it concisely. I'm sorry. Dig deeper! Han de Bruijn
From: Virgil on 16 Oct 2006 16:03 In article <1161008572.469763.93200(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > jpalecek(a)web.de schrieb: > > > mueckenh(a)rz.fh-augsburg.de napsal: > > > Dave L. Renfro schrieb: > > > > > > > Peter Webb wrote (in part): > > > > > > > > >> This is a complete red herring. There is no question that > > > > >> the Real generated by Cantor's proof is computable (r. e,) > > > > >> if the original list is, [...] > > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote (in part): > > > > > > > > > Of course. That's why the diagonal proof only proves the > > > > > existence of numbers which belong to a countable set i.e. the > > > > > set of constructible reals. This proof proves in essence that > > > > > the countable set of constructible real numbers is uncountable. > > > > > A fine result of set theory. > > > > > > > > You're overlooking Peter Webb's hypothesis "if the original > > > > list is". You need to have a list (x_1, x_2, x_3, ...) such > > > > that the function given by n --> x_n is computable. Thus, > > > > before you can conclude what you're saying (which sounds like > > > > a metalogic "proof by contradiction" to me, but no matter), > > > > you need to come up with a computable listing of the computable > > > > numbers (or at least, show that such a listing exists). > > > > > > One cannot compute a list of all computable numbers. By this > > > definition, > > > (1) the computable numbers are uncountable. > > > (2) There is no question, that the computable numbers form a countable > > > set. > > > This is a contradiction. It is not necessary to come up with a list of > > > all computable numbers. > > > > There is no contradiction. > > > > The fact that you cannot compute a list of all computable reals does > > not mean that there is no list of all computable numbers. There is one, > > > > and it is not computable. > > > The fact that you cannot compute a list of all reals does not mean that > there is no list of all reals. There is one, but it is not possible to > publish this list. But one can prove that no such "list" can exist.
From: mueckenh on 16 Oct 2006 16:04 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Virgil schrieb: > > > > > My sympathies to his poor students. > > > > I will tell them your ideas about the vase and then ask them about > > their opinion. But don't forget: They are not yet spoiled by what you > > call logic. > > > > > > > > - plainly cannot > > > > > comprehend the difference that swapping quantifiers makes. He cannot > > > > > comprehend that there might be a difference between the significance of > > > > > "every" in "Every girl in the village has a lover" and "John makes love > > > > > to every girl in the village". > > > > > > > > Is the Imaginator too simple minded to understand, or is it just an > > > > insult? The quantifier interchange is impossible in general, but it is > > > > possile for special *linear* sets in case of *finite* elements. > > > > > > For example? > > > > > > Does "Mueckenh" claim that, say, > > > "For every natural n there is a natural m such that m > n" > > > and > > > "There is a natural m, such that for every natural n, m > n" > > > are logically equivalent? > > > > > > All the elements are finite and linearly ordered. > > > > The second statement is obviously wrong, because there cannot be a > > natural larger than any natural. > > The quantifier exchange however is possible for sets of finite numbers > > n the following form: > > "For every natural n there is a natural m such that m >= n" > > and > > "There is a natural m, such that for every natural n, m >= n" > > This natural m is not fixed. It is the largest member of the set > > actually considered. > > Please let people know when you are not using standard terminology and > when you do this, please define your terms. What does it mean to say a > natural number "is not fixed"? One cannot know it, cannot call it by its name, but it is provably present. You should be familiar with this from of existence. It is like the well-order of the reals: present but very. Regards, WM
From: Virgil on 16 Oct 2006 16:05
In article <1eabb$453398c7$82a1e228$10949(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > stephen(a)nomail.com wrote: > > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > >>Is it possible to live in a (physical) world that is inconsistent? > > > > Perhaps. How could we know? > > How can we know, heh? Can things in the real world be true AND false > (: definition of inconsistency) at the same time? It would appear that in the world of physics things can be true one day and false the next and then true again the day after. > > Han de Bruijn |