In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 10 Apr., 23:24, Virgil <vir...@ligriv.com> wrote: > > > > > Many set theorists do so when claiming that an irrational > > > number in decimal representation "is more" than all its finite initial > > > segments > > > > It is certainly different from all its finite initial segments, and, at > > least in the sense of being longer is certainly "more". > > Then we can remove from it all FISs and yet have something left. What > is it?
WRONG! AGAIN!! AS USUAL!!!
While we can remove any one FIS and have something left, or even any finite set of FISs and have something left, note that since every member is in some FIS, removing all FISs removes all members too. > > > > WM is again ignoring the fact that, for a strictly increasing infinite > > sequence of any sort, the limit, of one exists, cannot be a member of > > the sequence. > > No, that is just my point. Infinitely many attempts to write > infinitely many 1's will fail, as the sequence > 0.1 > 0.11 > 0.111 > ... > shows. Infinitely many attempts end with a last 1 at a finite > position.
If one made infinitely many attempts there would not be a "last" position.
> Therefore 0.111... does not exist in its complete form. > Therefore it cannot be subject to digit- substitution.
If there is a position which does not exist, there must be a first one (or a last position that does exist), since positions form a well ordered set. >
> > > The limit is not suitable for Cantor's argument. He distinguished only > digits that belong to FISs. Compare the fact that every term of the > sequence > 0.1 > 0.11 > 0.111 > ... > has infinitely many zeros, but the limit has none.
I do not see any zeroes other than those extraneous ones before the radix point. > > So if Cantor substututes always the lfirst 0 by 2, he fails to change > anything in the limit 0.111... = 1/9.
Substituting 2 for the first 0 results in 2.111.... --