On 3/25/2013 7:07 AM, WM wrote: > 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. >
That is why 'all' means something different from 'one'.
That is the genius of what happened in the nineteenth century. Mathematics was seen to be a form of logic that could refer to parts of individuals as individuals in contrast to the classical logic that preceded that period.
>> 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.