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. ]
