On 25 Mrz., 00:49, Virgil <vir...@ligriv.com> wrote: > In article > <39dd320b-1f56-4cf7-bb03-f0f634420...@l5g2000yqe.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 24 Mrz., 20:39, Virgil <vir...@ligriv.com> wrote: > > > > 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. > > > We know your statements of faith. But where do you get aleph_0 lines > > without using lines of the infinite set of aleph_0 lines that, as > > provable in mathematics, are not sufficient? > > Which infinite sets of lines does WM claim are provably not sufficient?
All FISONs are not sufficient, because forall F in the set of FISONs: There are infinitely many natural numbers not covered by F and all its predecessors and all its followers.
> THEOREM: To have a subset of the infinite set of lines(FISONs) whose > union is |N, it is both necessary and sufficient that that subset of > lines also be infinite.
Nonsense. All FISONs cannot be sufficient, since no FISON is necessary.
Corollary: To catch a unicorn it is both necessary and sufficient to ask an infinity of horses to help. > > This theorem is valid
and its corollary is suitable, to show the strenght of ZFC.