Why can no one in sci.math understand my simple point?
in [Logic]
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From: WM on 16 Jun 2010 04:51 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. Nevertheless: Every number that can be determined, i.e., that is a number, belongs to a countable set. Regards, WM
From: WM on 16 Jun 2010 05:03 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). 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
From: WM on 16 Jun 2010 05:06 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. It is a form of a function or of a sequence. Regards, WM
From: WM on 16 Jun 2010 05:11 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. Regards, WM
From: |-|ercules on 16 Jun 2010 05:48
"WM" <mueckenh(a)rz.fh-augsburg.de> wrote ... > > By induction we prove: There is no initial segment of the (ANTI)diagonal > that is not as a line in the list. Right, therefore the anti-diagonal does not contain any pattern of digits that are not computable. Cantor's trick 123 456 789 Diag = 159 AntiDiag = 260 never produces a new sequence of digits. Herc |