In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> > Please in a clear in precise manner show that > > (A) No line contains all naturals. > > Implies > > (B) exist j, k, m, n : m e s_j & ~(m e s_k) & ~(n e s_j) & n e s_k. > > There is at least one line, that is lacking at least one natural, ok? > But if this were in another line containing all of the first one, the > other line would contain all naturals, ok?
Not at all. "At least one" does not mean "no more then one" so one has infinitely many different lines each of which is missing at least one natural, and with each actually be missing infinitely many, and all but finitely many, naturals. The collection of all FISONs (Finite Initial Sets Of Natuals) form such a family of sets.
> > At least one natural must be missing in every line. Indeed, there are > > infinitely many of em missing from every line. > > Then there are not all in the lines.
Nope, that doesn't work either, because with infinitely many lines in the form of FISONs, no natural is missing from more than finitely many of those infinitely many lines.