On 21 Mrz., 03:41, Virgil <vir...@ligriv.com> wrote:
> > > A needed number of lines has a first element. > > Nope! > > Any sufficient set of lines will have a first element, but as there are > pairs of such sufficient sets which are disjoint, there is no first > element common to all sufficient sets.
I am not interested in sufficient sets but in necessary sets. > > > If lines were chosen, that are superfluous, they cannot belong to the > > needed number. > > The number of lines needed is infinite, but there are infinitely many > pairwise disjoint sets
each of which contains only unnecessary lines. At least hitherto nobody has succeeded in naming a necessary line. Every named needed set starts with a not needed number.
> If the sets are pairwise disjoint, then
then the intersection of them is empty. How can they all contain the complete set of natural numbers in this case? And if at least two of these sets contain |N, how can they be disjoint? > > > > > > A set of needed lines exists. > > > Name the frist line. Or confess that not every set of natural numbers > > is a set if natural numbers that obeys the rules established in set > > theory for sets of natural numbers. > > That first sentence is not English. > > That second sentence does not scan in ENglish eoither..
What is eoither? Here my dictionary fails. > > > > > > A set of necessary lines does not.- > > > Ah, a needed line is not a necessary line? > > No particular line is needed as long as some subsequent line is included.
What is the first one? I can prove by induction, that there is no necessary subsequent line, except a / the last one.
It is a pity. You and your ilk are unable to apply induction, but boast of of "transfinite induction". Note: Transfinite induction can be applied to show that its applyer is a fool that should be inside a mad-house. Further merits of transfinite induction are not known.