On 21 Mrz., 08:57, William Hughes <wpihug...@gmail.com> wrote: > On Mar 21, 8:46 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > My question remains: What is the subset of necessary lines? > > There is no such thing as a necessary line.
Correct. That is because the asserted aim cannot be established. In order to reach a goal that is impossible to reach, no attempt is necessary.
> There is such a thing as a sufficient set of > lines (all sufficient sets are composed > entirely of unnecessary lines, which means > that you can remove any finite set of lines
Why only finite sets? What property is changed if infinitely many are there? If there are infinitely many unnecessary lines, they all can be removed - by their property of being unnecessary. Otherwise there must be some necessary line among them.
You try to cheat: If we have infinitely many unnecessary lines, then some of them become necessary. But of course, we cannot find a first one of this miraculous set. Do they all remain sets of natural numbers? Or will also this property transform? And why?
> from a sufficient set and get a different sufficient > set of lines).
You claim that there is a sufficient set, but every line you have offer ed up to now is not necessary and not sufficient. Amazing that you want to sell that as mathematics.
> The intersection of all sufficient > sets is empty.
Correct. That is because the sets are only believed to be sufficient, but in mathematics in fact, they are not sufficient.
Try to answer this question to yourself: Why do you claim that an infinite set of unnecessary lines must not be removed completely, i.e., contains lines that must remain, but an infinite set of natural numbers must contain only natural numbers?