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? > > 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. Therefore 0.111... does not exist in its complete form. Therefore it cannot be subject to digit- substitution.
> It is more than any FIS, by the property that it is the limit of a > sequence of FISs. > 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.
So if Cantor substututes always the lfirst 0 by 2, he fails to change anything in the limit 0.111... = 1/9.