Why can no one in sci.math understand my simple point?
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From: Virgil on 16 Jun 2010 18:26 In article <67281815-918f-4f3d-85dd-674f22e6fb31(a)c10g2000yqi.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > Reason: Cantor either proves that a countable set is uncountable or > that a constructible/computable/definable number is not constructible/ > computable/definable. Cantor does non of those things. WM tries to but does not succeed.
From: |-|ercules on 16 Jun 2010 18:44 "WM" <mueckenh(a)rz.fh-augsburg.de> wrote > 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. Sorry I thought you meant the anti-diagonal is computed to every finite prefix as this is a more direct contradiction to any new sequences of digits being possible. Herc
From: |-|ercules on 16 Jun 2010 18:53 "WM" <mueckenh(a)rz.fh-augsburg.de> wrote > 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.- It's darn well more logical definition than your superinfinity based on your circular reasoning "no box contains the box numbers that don't contain their own box number". Oh but you have a backup proof, this is a new sequence because we *construct* it like so: CANTORS PROOF Defn: digit 1 is different, and digit 2 is different, digit 3 is different, ... Proof: digit 1 is different, and digit 2 is different, digit 3 is different... Therefore it's a different number! > > 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 We invented natural numbers, partitioned the space between natural numbers recursively with one of ten options. It's ridiculous to define a different number as "the other nine options ad infinitum". Herc
From: Tim Little on 16 Jun 2010 23:26 On 2010-06-15, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > No. You cannot form a list of all computable Reals. If you could do > this, then you could use a diagonal argument to construct a > computable Real not in the list. You can form a list of all computable reals (in the sense of mathematical existence). However, such a list is not itself computable. - Tim
From: Peter Webb on 16 Jun 2010 23:25
> > It is proof that there is no countable set of all real numbers, since > any alleged such set is provably and constructably incomplete. > > Similarly, it is proof that there is no countable set of all > constructable numbers, since any alleged such set is provably and > constructably incomplete. I hate to disagree with you, because we are on much the same "side", but this is not correct. Cantor's proof shows that you cannot form a list of all Reals. This is not the same as the Reals being uncountable. You can use Cantor's diagonal construction to similarly prove that you cannot form a list of all computable numbers. However the computable numbers are in fact countable. You can't simply equate the two concepts; they are not exactly the same thing. |