On 15 Mrz., 17:13, William Hughes <wpihug...@gmail.com> 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.
Let's first prove that already two cannot be necessary by the fact that two always can be replaced by one of them without changing the contents. Then it is clear that two or more cannot be necessary and from this immediately follows that they also cannot be sufficient. > > Two lines can never do a better job than 1. > > Any finite number of lines is necessary but not sufficient.
Wrong. Why do you resist to apply logic? > > Any finite number of lines can never do a better job than 1. > > An infinite number of lines is necessary and sufficient
Exercise: If of any two line one is not necessary, how many of infinitely many lines are not necessary? > > An infinite number of lines can do a better job than 1
That is a confession of irrational belief. With exactly the same right you could state: An infinite number of even naturals contains an odd natural.
You may claim so, but it is not part of mathematics. You should accept: If someone claims infinity and is not even able to show two, then his claim is nothing that could be provable, hence nothing that belongs to mathematics. > > [In potential infinity things go > > Any number of findable lines is not sufficient > > An unfindable line is necessary and sufficient
In any case the last line contains every number of the list. This is so by construction. We have the choice between 1 line (in potential infinity) and 0 lines (in actual infinity).