Cantor Confusion
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From: jpalecek on 17 Oct 2006 08:46 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > > 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. > > > > > > > > Nope. You are mixing two approaches.. > > > > > > > > A set X is countable if there exists a surjective function,f, > > > > from the natural numbers to X > > > > > > > > There are two possibilities > > > > > > No. > > > > > > > > A: you allow arbitrary functions f > > > > > > > > B: you allow only computable functions f > > > > > > What is a function which is not computable? > > > > > > > > In case A, the computable reals are countable [ but you > > > > also have arbitrary reals, and the set of reals (computable > > > > and arbitrary) is uncountable.] > > > > > > > > In case B. the computable reals are not countable > > > > (i.e. there is no list of computable reals) > > > > > > > > You cannot arrive at a contradiction by taking one result from > > > > case B (the computable reals are uncountable) > > > > and one result from case A > > > > (the computable reals are countable). > > > > > > I take not at all results from your mysterious cases. > > > I see: Every list of computable numbers supplies a diagonal number > > > which is computable but not contained in the list. > > > And I see: Every list of real numbers supplies a diagonal number which > > > is not contained in the list. > > > There is absolutely no difference. > > > > > > > If real numbers are also computable numbers there is absoluetly > > no difference. If real numbers contain both computable > > real numbers and non-computable real numbers then > > there is a difference. But in either case > > > > Every list of real numbers supplies a diagonal number which > > is not contained in the list. > > > > so any way you cut it there is no complete list of real numbers. > > And there is no complete list of computable numbers. But they are > countable. Hence the diagonal argument does not prove anything. This is nonsense. You say "my proof is flawed, therefore your proof is flawed". You'd better think more about your proof.
From: Han de Bruijn on 17 Oct 2006 08:47 David Marcus wrote: > cbrown(a)cbrownsystems.com wrote: > >>Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >>>I talked about the real world, physics as an empirical science, >>>not about artifical theoretical constructs. In the real world, >>>Schrodinger's cat is dead :-( >> >>I thought you kept up with physics? >> >>http://physicsweb.org/articles/news/4/7/2 >> >>The device is conducting electricity in a clockwise fashion; and the >>device is not conducting electricity in a clockwise fashion. > > That interpretation of the experiment is probably dependent on theory. > Try this: > > http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/bmstartE.htm David Marcus is an adherent of some rather outdated Quantum Mechanical theories, as have been proposed in the middle of the past century, by David Bohm. Especially Bohm's theory of "hidden variables", which have never been found. (And IMO will never be found) Han de Bruijn
From: William Hughes on 17 Oct 2006 09:00 mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > > > 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. > > But the function of balls/numbers removed from the vase is a > continuously (stepwise) increasing one, containing all natural numbers > at noon? > Nope. Let, g(t) (t < noon) be the "function of balls/numbers removed from the vase". Then, for all (t<noon) g(t) is an integer. So g(t) is not a set of natural numbers, so to say that g(t) contains all natural numbers is nonsense. So g(t) is an increasing function g(t) is continuous (i.e. its jumps are limited to 1) g(noon) does not contain all natural numbers - William Hughes
From: Han de Bruijn on 17 Oct 2006 09:06 stephen(a)nomail.com wrote: > Anyway, how NOT EXACTLY IS the world? Is there an aether? > Do photons have mass? Share your secrets. http://hdebruijn.soo.dto.tudelft.nl/QED/ Han de Bruijn
From: Han de Bruijn on 17 Oct 2006 09:09
stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>stephen(a)nomail.com wrote: > >>>You have not answered the question about how one determines >>>if a thing in the real world is true. I can guess you >>>will say something about measurements, but how does one >>>know that your measurements are "true", or that they truly >>>correspond to "a thing in the real world", and so on. >>>It is a big ugly kettle of philosophical fish. > >>Not only philosophical fish. Also religious fish. And political fish. >>And scientific fish. Actually everyday's life fish. You are right! > >>>I agree that it is sensible to assume that the Universe >>>is consistent, but given how strange and unintuitive >>>the Universe can be, who knows. > >>I think we agree on the above. But it doesn't mean that we cannot answer >>_part_ of the question: do INFINITIES exist or not. Are they true or are >>they false? And IMO _that_ can be decided _now_, without rocket science. > > No, I do not see how we can decide that. We cannot observe > infinities, but that does not mean they do not exist. Unless > of course that is some axiom of yours, but again, you cannot > know that that axiom is "true". > > There are people who are convinced that the Universe is truly > eternal, that is always has been, and always will be. I suppose > their arguments have convinced them that an infinity does exist. > What irrefutable evidence do you have that the Universe has > not always existed? The scientific attitude: thy shall not believe what thy cannot measure. Han de Bruijn |