Cantor Confusion
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From: William Hughes on 16 Oct 2006 10:43 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. Yes it does if the only type of list is one you can compute. And if the only type of reals that exist are the reals you can compute then the only type of lists that exist are the lists you can compute (Consider the list of the decimal digits of a real. This list is computable iff the real number is computable. So if non-computable lists exists, so do non-computable reals. - William Hughes
From: imaginatorium on 16 Oct 2006 10:43 mueckenh(a)rz.fh-augsburg.de wrote: > 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? By the application of a little elementary mathematical knowledge, I should think. (1) For any positive value nu (integer, but actually a real will do too), for a sufficiently small value of tau, the number of balls in the vase at time noon-tau is greater than nu. This can be derived tediously, but obviously, from the set of step functions, one for each natural n, representing the state of the nth ball. Therefore the limit to the number of balls as time approaches noon from the "left" diverges. (2) For the value t=noon, there does not exist any n for which the step function representing the state of ball n has the value IN. Therefore the set of balls with the corresponding value IN is empty, and the number of balls is zero. Therefore the limit as t->noon does not equal the value at noon. This simply follows from the statements in the problem. Brian Chandler http://imaginatorium.org
From: stephen on 16 Oct 2006 11:19 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: >> >>>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 What does it mean for a thing in the real world to be true? How do you know if a thing in the real world is true? Consider the twin slit experiment. Is the fact that none of the following accurately describe the situation an inconsistency? a) the photon goes through one slit b) the photon goes through both slits c) the photon goes through neither slit Stephen
From: Randy Poe on 16 Oct 2006 11:20 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > According to the ZFC system: The vase is empty at noon, because all > > > natural numbers left it before noon. > > > By means of the ZFC system we can formulate sequences and their limits > > > in mathematical language. From this it follows that lim {n-->oo} n > 1. > > > And from this it follows that the vase is not empty at noon. > > > > By what axiom do you conclude that the limit as t increases towards noon > > of any function and the value of that function at noon must be the same? > > By that or those axiom(s) which lead(s) to the result lim {t-->oo} 1/t > = 0. Here is your theorem: Let f(x) be any function f:R->R. Then lim(x->0-) f(x) = f(0). That is, the limit of f(x) as x approaches 0 from the left is f(0). Can you show me how the axiom(s) you describe prove that theorem? Can you then show me how the theorem applies to this function? f(x) = 1 if x<0, f(x) = -1 if x>=0. Thanks. - Randy
From: MoeBlee on 16 Oct 2006 13:20
mueckenh(a)rz.fh-augsburg.de wrote: > A good, if no the best source to learn about the different meanings of > infinity would be Cantor's collected works. Set theory has advanced since Cantor. The best source to learn about current set theoretic definitions of 'infinite' is not Cantor. MoeBlee |