In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
How is it that in WMytheology the set of all lines does not contain somewhere every member of every line?
> > > 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?
He did not say only finite sets of lines. And it possible from any infinite set of lines to remove all but infinitely many, which does allow removal of infinitely many.
> If there are infinitely many unnecessary lines, they all can be > removed -
Only in Wolkenmuekenheim.
> > You try to cheat
WM actually does cheat, but does it so badly, he almost alwasys gets caught at it. > > > from a sufficient set and get a different sufficient > > set of lines). > > You claim that there is a sufficient set
The set of all lines is obviously sufficient to cover all naturals, but is equally obviously not necessary, as any one line can be removed without uncovering any natural.
In fact, from any set of lines that is sufficient, any one line can be removed and the set of remaining lines will still be sufficient, but if one removes infinitely many lines what remains need not be sufficient.
> 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.
It is quite good asmathematics, it is only bad as WMytheology.
> Try to answer this question to yourself: Why do you claim that an > infinite set of unnecessary lines must not be removed completely
I don't claim that.
Outside of Wolkenmuekenheim, from any set of lines sufficient to cover all naturals SOME, but not all, infinite sets of lines may be removed and still leave all naturals covered.
WM claims to know how to map bijectively the set of infinite binary sequences, B, linearly to the set of reals and then map that image set of reals linearly ONTO the set of all paths, P, of a Complete Infinite Binary Tree.
But each binary rational in |R is necessarily the image of two sequences in B but that one rational can then only produce one image in P, so the mapping cannot be the bijection WM claims.
SO that WM is, as usual with things mathematical, wrong. --