In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> It is easy. Look at the sequence > 0.1 > 0.11 > 0.111 > ... > Each term has only natural indices. > The limit 0.111... has not only natural indices, because all natural > indices are in the terms of the sequence. So no index or set of > indices remains to distinguish 0.111... from all terms of the > sequence.
While it is true that no single index or even finite set of indices is enough, the infinite set of all indices, being larger than any FISON, is quite enough to distinguish the limit from any term of the sequence.
Note that outside of Wolkenmuekenheim, such infinite sets as the set of all natural numbers, the set of all integers, the set of all rationals and the set of all reals (the last being uncountable) are quite standard.
If WM wishes to create a world of his own where such standard usages are exiled, he is quite free to do so, but cannot expect the vast majority of mathematicians to follow him into his tiny cramped corrupted world. --