In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 4 Mai, 18:30, Dan <dan.ms.ch...@gmail.com> wrote: > > > > Either they're both relevant to Cantor's argument, or they're both > > > > irrelevant . > > > > > Cantor's argument IS valid! > > > > Why should it be considered important under the definition of > > equality? If they have at least one different digit , THEY'RE > > DIFFERENT. > > But there always only finitely many with at least one different digit > whereas there are always infinitely many with none different digit.
Name one! > > > Every finite digit appears in the list at least once , but NEVER DO > > ALL OF THEM APPEAR AT ONCE , IN THE SAME NUMBER. > > Stop shouting.
Since nothing less seems able to penetrate the crenelated battlements of Schloss Wolkenmuekenheim, why not? > > > > > THIS IS REQUIRED TO DISPROVE CANTOR : > > exists n , forall m , a_nm = d_m > > Of course, in a list that contains all rationals, this is easily > proved. > A simpler case is the list: > > 0.0 > 0.1 > 0.11 > 0.111 > ...
With 0,0111... as its anti-dialgonal, so WM loses again.
> Your assertion that in your anti-diagonal there are more than all > digits that are already covered by the list is simply nonsense.
Since no one made any such argument, except WM, it is WM's nonsense.
The standard mathematical definition of a set, S, being countable is that there is some mapping from |N to S that is surjective, i.e., such that every member of S is the image of some member of |N.
Such a mapping constitutes a 'listing' of the members of S with the members position(s) in the list being the appropriate member(s) of |N.
Thus for WM, or anyone else, to successfully claim that the set of reals is countable is equivalent to claiming that the set of real is listable.
Thus (reals countable) <==> (reals listable).
But Cantor has shown that there is no complete list of real possible as every listing of reals necessarily omits some reals.
> > > My list is constructed such that all possible squences of digits 1 are > already in the list. No longer sequence is possible (otherwise it > would be in the list) and a shorter sequence cannot support Cantor's > claim.
Unless your list claims to list all reals, it has no effect on the validity of the Cantor diagonal argument, which remains valid in spite of it. > --