Why can no one in sci.math understand my simple point?
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From: Virgil on 16 Jun 2010 15:44 In article <995d761a-f70f-4bca-b961-8db8e1663e3f(a)d37g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 15 Jun., 22:24, Virgil <Vir...(a)home.esc> wrote: > > > > �Note that it is possible to have an uncomputable number whose decimal > > expansion has infinitely many known places, so long as it has at least > > one unknown place. > > That is mathematically wrong. It may not match every definition of 'uncomputable', but otherwise it is right. > Nevertheless: Every number that can be determined, i.e., that is a > number, belongs to a countable set. If every set of numbers is provably countable, that means that for every set there is a constructable surjection from N to that set, and for any such surjection, Cantor proves there is a real number not covered. So one wonders what WM's definition of countability is? onte that as soon as one has the standard N and powersets, WM loses.
From: Virgil on 16 Jun 2010 15:49 In article <35ea24dc-1786-4dad-8c64-d8011ea2594c(a)x21g2000yqa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 15 Jun., 22:45, Virgil <Vir...(a)home.esc> wrote: > > In article > > <f78b53d6-24d1-42e2-86bd-1dd0893b8...(a)q12g2000yqj.googlegroups.com>, > > > > > > > > > > > > �WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 15 Jun., 16:06, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > WM says... > > > > > > >On 15 Jun., 12:26, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > > > >> (B) There exists a real number r, > > > > >> Forall computable reals r', > > > > >> there exists a natural number n > > > > >> such that r' and r disagree at the nth decimal place. > > > > > > >In what form does r exist, unless it is computable too? > > > > > > r is computable *relative* to the list L of all computable reals. > > > > That is, there is an algorithm which, given an enumeration of computable > > > > reals, returns a real that is not on that list. > > > > > > In the theory of Turing machines, one can formalize the notion > > > > of computability relative to an "oracle", where the oracle is an > > > > infinite tape representing a possibly noncomputable function of > > > > the naturals. > > > > > We should not use oracles in mathematics. > > > > WM would prohibit others from doing precisely what he does himself so > > often? > > > > > A real is computable or not. My list contains all computable numbers: > > > > > 0 > > > 1 > > > 00 > > > ... > > > > > This list can be enumerated and then contains all computable reals. > > > > If that list is .0, .1, .00, ..., then it contains no naturals greater > > than 1. > > This list is the list of all words possible in any language based upon > any finite alphabet. The list is given in binary. All alphabets, all > languages and all dictionaries are contained in later, rather long but > finite lines. > Therefore this is a list of everything (that can meaningfully be > expressed). Such a list never ends. > > This list does not allow for a diagonal, because that is a meaningless > concept. (That is proved in my list, in a later line.) > > Regards, WM Is that like the typing monkeys eventually producing "Hamlet"? Note that, unless there is a cap on the length of what is acceptable as a "word", or some other restrictionon what are allowed to be words, there can be no limit to the number of possible words
From: Virgil on 16 Jun 2010 15:53 In article <250e1f42-8d35-42f0-969b-3f919b4ce5e4(a)c33g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 15 Jun., 22:52, Virgil <Vir...(a)home.esc> wrote: > > In article > > <4b892c9b-5125-46b6-8136-4178f0aca...(a)b35g2000yqi.googlegroups.com>, > > > > �WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 15 Jun., 16:17, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > > > > > In this sense, the antidiagonal of the list of all computable reals > > > > is definable (but not computable). > > > > > That is nonsense. To define means to let someone know the defined. If > > > he knows it, then he can compute it. > > > > There are undecidable propositions in mathematics, so if P is one of > > them then "x = 1 if P is true otherwise x = 0" defines an uncomputable > > number. > > 3 is a number. n is a number form. > 5 < 7 is an expression. m < n is the form of an expression. > f(n) = (1 if Goldbach is correct) is not a function and it is not > computable. For a long time, g(n) = ( if FLT is true then 1 else 0) was unknown, but now it is known. So unless WM can prove that Goldbach's conjcture will be forever undetermined, he cannot claim f(n) = (1 if Goldbach is correct, 0 otherwise) is not a function.
From: Virgil on 16 Jun 2010 15:57 In article <b4412a8d-c10e-481a-89dc-a7ffa672f3ba(a)z10g2000yqb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 00:13, "K_h" <KHol...(a)SX729.com> wrote: > > > > No one item on the list contains pi in its entirety. > > > > True, there is no entry for pi, in its entirety, on the list > > but all of the digits of pi are there along the diagonal. > > By induction we prove: There is no initial segment of the diagonal > that is not as a line in the list. And there is no part of the > diagonal that is not in one single line of the list. Which, while true, is irrelevant to any of the issues under discussion. If one has any function from N ONto an arbitrary set, S, of real numbers, then there are countably many real numbers not members of S which can be constructed from the function.
From: Virgil on 16 Jun 2010 16:04
In article <9df240be-eaec-4d46-bd74-42868f4970ec(a)g19g2000yqc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 02:39, "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> > wrote: > > > Nevertheless your "definition" belongs to a countable set, hence it is > > > no example to save Cantors "proof". > > > > > Either all entries of the lines of the list are defined and the > > > diagonal is defined (in the same language) too. > > > > Yes. If you provide a list of Reals, then the diagonal is computable and > > does not appear on the list. > > Delicious. Cantor shows that the countable set of computable reals is > uncountable. That would require that one can have a list of all and only the computable numbers which is already known to be impossible. So WM is wrong again, as usual. |