"WM" <mueckenh@rz.fh-augsburg.de> wrote > On 16 Jun., 14:19, "J. Clarke" <jclarke.use...@cox.net> wrote: >> On 6/15/2010 4:33 PM, WM wrote: >> >> >> >> >> >> > On 15 Jun., 21:38, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: >> >> WM says... >> >> >>> On 15 Jun., 18:53, stevendaryl3...@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".
