Why can no one in sci.math understand my simple point?
in [Logic]
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From: Daryl McCullough on 15 Jun 2010 12:53 WM says... > >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. That's just not true. 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. This is very basic stuff. I'm a little surprised that you are so unfamiliar with it. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 15 Jun 2010 12:57 WM says... > >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. On the contrary! Many real numbers in physics are not computable to infinite precision (for example, the fine structure constant). Yet, we can certainly compute other real numbers *relative* to such parameters. We can easily devise an algorithm to compute the square of the fine-structure constant, for example. This algorithm will take as an input an approximation to the fine-structure constant, and will return an approximation to the square of the fine-structure constant. In this sense, the square of any real number is computable relative to that real number. -- Daryl McCullough Ithaca, NY
From: Ronald Bruck on 15 Jun 2010 13:01 In general, when you have to write the statement in the Subject: header, it's time to pause and reflect. -- Ron Bruck
From: WM on 15 Jun 2010 13:27 On 15 Jun., 17:59, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > WM <mueck...(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. A real number cannot be defined other than by a finite definition. All are in the list. Have you ever obtained a real number from an infinite definition or sequence? When was that? Who told you? What number was it? (Please give it by finite definition, otherwise the internet will break down.) Regards, WM
From: Jesse F. Hughes on 15 Jun 2010 13:27
WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 15 Jun., 16:23, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> writes: >> > So (B) is equivalent to the statement "there exists an uncomputable >> > number". >> >> Right. But why then did you say the number was computable? > > And in what form does it exist? This question seems utterly meaningless. In what form does the barbecue pork bun I'm eating exist? -- Jesse F. Hughes "So far as this negative attitude toward life is concerned, Buddhism is merely Taoism a little touched in its wits." -- Lin Yutang, /My Country and My People/ |