"David R Tribble" <email@example.com> wrote in message news:firstname.lastname@example.org... > WM wrote: > >>... consider the list of finite initial segments of natural numbers > > 1 > > 1, 2 > > 1, 2, 3 > > ... > > > > According to set theory it contains all aleph_0 natural numbers in its > > lines. But is does not contain a line containing all natural numbers. > > Therefore it must be claimed that more than one line is required to > > contain all natural numbers. This means at least two line are > > necessary. > > That is correct. In fact, all Aleph_0 lines are required > (necessary sufficient) to contain all of the naturals.
This is sufficient but not necessary. (Aleph_0 lines are necessary and sufficient.)