In article <email@example.com>, William Hughes <firstname.lastname@example.org> wrote:
> On Mar 14, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > <snip> > > >... 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. There are no special lines necessary, but there must be at > > least two. In this case, however, we can prove, by the construction of > > the list, that every union of a pair of lines is contained in one of > > the lines. This contradicts the assertion that all natural numbers > > exist and are in lines of the list. > > > Nope. > > Nope, two lines are necessary but not sufficient. > > Two lines can never do a better job than 1. > > Any finite number of lines is necessary but not sufficient. > > Any finite number of lines can never do a better job than 1. > > An infinite number of lines is necessary and sufficient > > An infinite number of lines can do a better job than 1 > > [In potential infinity things go > > Any number of findable lines is not sufficient > > An unfindable line is necessary and sufficient > > An unfindable line can do a better job than > any number of findable lines.