In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 23 Mrz., 18:48, David R Tribble <da...@tribble.com> wrote: > > 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. > > That claim is easily disproved. > For all n in N: The first n lines are neither necessary nor sufficient > to contain all of the naturals. > . > > Regards, WM
A red letter day, WM is finally RIGHT about something!
But if David had left out the world "all", and said merely "In fact, Aleph_0 lines are required (necessary sufficient) to contain all of the naturals." then David would have been correct, since EVERY set of aleph_0 lines is sufficient but no set of less than aleph_0 lines is sufficient. --