In article <email@example.com>, firstname.lastname@example.org wrote:
> >Consider a Cantor-list that contains a complete sequence (q_k) of all > >rational numbers q_k. The first n digits of the anti-diagonal d are > >d_1, d_2, d_3, ..., d_n. It can be shown *for every n* that the > >Cantor- > >list beyond line n contains infinitely many rational numbers q_k that > >have the same sequence of first n digits as the anti-diagonal d. > > > >Proof: There are infinitely many rationals q_k with this property. > >All > >are in the list by definition. At most n of them are in the first n > >lines of the list. Infinitely many must exist in the remaining part > >of > >the list. So we have obtained: > > > >For all n exists k: d_1, d_2, d_3, ..., d_n = q_k1, q_k2, q_k3, ..., > >q_kn. > >This theorem it is not less important than Cantor's theorem: For all > >k: d =/= q_k. > > > >Both theorems contradict each other with the result that finished > >infinity as presumed for transfinite set theory is not a valid > >mathematical notion.
Outside or Wolkenmuekenheim it is of no importance at all, since what WM claims follows from it does not follow from it.
A list of all rationals between 0 and 1 need not contain any non-rationals, and cannot contain all irrationals, of which there are at least as many as there are rationals, so is clearly an incomplete listing of all the numbers between 0 and 1, just as Cantor said! --