Why can no one in sci.math understand my simple point?
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From: WM on 16 Jun 2010 07:06 On 16 Jun., 03:05, "Mike Terry" <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > "Virgil" <Vir...(a)home.esc> wrote in message > > news:Virgil-3B2F0B.16291815062010(a)bignews.usenetmonster.com... > > > In article <87sk4ohwbt....(a)dialatheia.truth.invalid>, > > Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > Virgil <Vir...(a)home.esc> writes: > > > > > 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. > > > > You need infinitely many unknown places. > > > If the value of some decimal digit of a number depends on the truth of > > an undecidable proposition, can such a number be computable? > > Yes - e.g. imagine just the first digit of the following number depends on > an undecidable proposition: > > 0.x000000000... > > There are only 10 possibilities for the number, and in each case it is > obviously computable... These numbers are computable. What you wrote, however, is not a number but a form of a number. 3 > 1 is a correct proposition. x > 1 is the form of a proposition. But all that is not of interest for the present problem: All definable, computable, and somehow identifiable numbers and forms of numbers are within a countable set. Therefore Cantor proves the uncountability of this countable set. Regards, WM
From: J. Clarke on 16 Jun 2010 08:19 On 6/15/2010 4:33 PM, WM wrote: > On 15 Jun., 21:38, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> WM says... >> >> >> >>> On 15 Jun., 18:53, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >>>> For example, we can define a real r as follows: >> >>>> r = sum from n=0 to infinity of H(n) 2^{-n} >> >>>> where H(n) = 1 if Turing machine number n halts on input n, >>>> H(n) = 0 otherwise. >> >>>> That's definable, but it is not computable. >> >>> Anyhow it is not a definition. >> >> It certainly is. It uniquely characterizes a real number, >> so it's a definition. > > It does not. If it would, the number could be computed. > Who defines what Turing machine number n would do? Can you say "circular argument"? It's not a number because it's not computable and that proves that all numbers are computable.
From: J. Clarke on 16 Jun 2010 08:05 On 6/15/2010 11:59 AM, Aatu Koskensilta wrote: > WM<mueckenh(a)rz.fh-augsburg.de> writes: > >> We should not use oracles in mathematics. > > Nonsense. Using orcales we can show for example that the P = NP problem > can't be solved using any technique that relativizes. This is a useful > result. > >> A real is computable or not. My list contains all computable numbers: >> >> 0 >> 1 >> 00 >> ... > > Your list doesn't appear to contain any real at all, just finite binary > sequences. Did someone redefine the real numbers to exclude all numbers that consist only of the digits 0 and 1?
From: WM on 16 Jun 2010 11:16 On 16 Jun., 11:48, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "WM" <mueck...(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. > Sorry, you misquoted me. I wrote: 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. But I have to excuse because I wrote somewhat unclear. The meaning is: 1) Every initial segment of the decimal expansion of pi is in at least one line of your list 3. 3.1 3.14 3.141 .... What we can finde in the diagonal (not the anti-diagonal), namely 3.141 and so on, exactly that can be found in one line. This is obvious by construction of the list. 2) Every part of the diagonal is in at least one line. That means, every part is in one single line, or there are parts that are in different lines but not in one and the same. The latter proposition can be excluded. If there are more than one lines that contain parts of pi, then it can be proved, be induction, that two of them contain the same as one of them. This can be extended to three lines and four lines and so on for every n lines. Hence we prove that all of pi, that is contained in at least one of the finite lines of your list, is contained in one single line. Conclusion: Either the complete diagonal pi does not exist, or it exists also in one and the same single line. As the latter is wrong, so is the former. There does not exist an actually infinite sequence. Actually infinite mean finished infinite. That is nonsense. (The proof has the same status as Cantor's diagonal proof. Also his proof is valid only for all finite n. In a similar way we find above: For all finite n: All of pi that is in the list up to line n, is in a single line of the list.) Regards, WM
From: WM on 16 Jun 2010 11:39
On 16 Jun., 14:19, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > On 6/15/2010 4:33 PM, WM wrote: > > > > > > > On 15 Jun., 21:38, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> WM says... > > >>> On 15 Jun., 18:53, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >>>> For example, we can define a real r as follows: > > >>>> r = sum from n=0 to infinity of H(n) 2^{-n} > > >>>> where H(n) = 1 if Turing machine number n halts on input n, > >>>> H(n) = 0 otherwise. > > >>>> That's definable, but it is not computable. > > >>> Anyhow it is not a definition. > > >> It certainly is. It uniquely characterizes a real number, > >> so it's a definition. > > > It does not. If it would, the number could be computed. > > Who defines what Turing machine number n would do? > > Can you say "circular argument"? It's not a number because it's not > computable and that proves that all numbers are computable.- To be computable can be use as a *definition* of number. What is a natural number that cannot be counted or used for counting? What is a name that cannot be named? (A stone remains a stone, even if nobody names or knows it, but a thought that remains unthought forever is not a thought.) A real number could also be called a computable entity. Then we would earlier have recognized the charlatanism implicit in uncomputable or undefinable real "numbers". Cantor himself did not share that idea. He was convinced that the number of definition is not countable. Otherwise he was too much inclined to real mathematics to have upheld the claim of an uncountable set of reals. Regards, WM |