Cantor Confusion
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From: mueckenh on 16 Oct 2006 10:05 Randy Poe schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Randy Poe schrieb: > > > > > Han de Bruijn wrote: > > > > > I merely note that there is no requirement in the problem that > > > > > the limit be the value at noon. > > > > > > > > The limit at noon - iff it existed - would be the value at noon. > > > > > > Wrong. That is a flat out incorrect statement showing a > > > fundamental misunderstanding about what limits mean. > > > > > > A CONTINUOUS function at x0 has the property that the > > > limit of f(x) as x->x0 is f(x0). But not all functions are > > > continuous. > > > > And you are in charge of determining which functions are continuous and > > which are not? > > Where do you get this stuff from? > > How do you translate a statement that some functions are not > continuous into "I am in charge of determining if some functions > are continuous"? Because it seems a bit mysterious, how you know or can define that the function of balls in the vase was not continuous. Or, may be, because you will accept that the function is continuous if the balls are taken out in the sequence 1, 11, 21, .... > > No, I am not in charge. Non-continuous functions are non-continuous > now and forever. They were non-continuous before I existed, they will > remain non-continuous after I'm gone. > > The number of balls in the vase is such a function. How do you acquire that knowledge? > > > The function f(t) = 9t is continuous, because the function 1/9t is > > continuous. > > Yes, but that is not the number of balls in the vase. > For the t-th transaction 9t is the number of balls in the vase. Regards, WM
From: mueckenh on 16 Oct 2006 10:16 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? Regards, WM
From: mueckenh on 16 Oct 2006 10:22 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. Every list of reals can be shown incomplete in exactly the same way as every list of computable reals can be shown incomplete. Regards, WM
From: William Hughes on 16 Oct 2006 10:29 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? > Nope. [There are no balls removed from the vase at noon] The function 9n has nothing to do with the number of balls in the vase at noon. - William Hughes
From: Han de Bruijn on 16 Oct 2006 10:35
stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>Dik T. Winter wrote: > >>>In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> >>>Han.deBruijn(a)DTO.TUDelft.NL writes: >>> > Virgil schreef: >>>... >>> > > I do not object to the constraints of the mathematics of physics when >>> > > doing physics, but why should I be so constrained when not doing physics? >>> > >>> > Because (empirical) physics is an absolute guarantee for consistency? >>> >>>Can you prove that? > >>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? Han de Bruijn |