In article <y-udnc3rbL5NOvrMnZ2dnUVZ_sqdnZ2d@giganews.com>, fom <fomJUNK@nyms.net> wrote:
> On 4/12/2013 12:57 AM, WM wrote: > > Matheology § 246 > > > > Cantor's list contains real numbers r as binary or decimal fractions. > > WM is wrong again. > > Cantor's list consists of one representation for > a real number that is not on the given list purported > to consist of representations for every number.
Actually the lists are not Cantor's, but are challenges to Cantor, each of which is shown by Cantor to be an incomplete listing of all binary sequnces of the letters "m" and "w". > > > Real numbers, however, are /limits/ of binary or decimal fractions. > > > > Yes. This is why the arithmetic of real numbers is not the > arithmetic of rational numbers, although the latter is > representable within the former. > > > For every terminating fraction of r, Cantor obtains a difference > > between r and the due terminating fraction of the anti-diagonal d: > > r_nn =/= d_n. > > He does not obtain one. He constructs one based on > syntactic criteria. > > > He concludes that this remains true for the limits of > > the list numbers r and d by using the argument: different sequences > > have different limits. But it is well known that this argument is not > > admissible in proofs because it is false. > > But, the argument is based on the representation > of real numbers with respect to representation > according to the output of the Euclidean algorithm. > > There is no assumption concerning the convergence > of partial sums whatsoever. > > If WM's statement were to be given credence, the Euclidean > algorithm of long division could no longer be considered > as providing a faithful representation of distinct real > numbers (or rational numbers for that matter). --