In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 5 Apr., 12:08, William Hughes <wpihug...@gmail.com> wrote: > > On Apr 5, 11:09 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 4 Apr., 23:08, William Hughes <wpihug...@gmail.com> wrote: > > > > > > Nope. Any single element can be removed. This does not > > > > mean the collection of all elements can be removed. > > > > > You conceded that any finite set of lines could be removed. What is > > > the set of lines that contains any finite set? Can it be finite? No. > > correct > > > So the set of lines that can be removed form an infinite set. > > > > More precisely. There is an infinite set of lines D > > such that any finite subset of D can be removed. > > What has to remain?
If one has any set, S, which is order isomorphic to the set of naturals with their natural well-ordering, one can form the family, F, of FISs of that set (finite initial segments).
Then any infinite subset of F will union to give the original S but no finite subset of F will union to give back S.
That WM seems incapable of comprehending this simple truth marks his as mathematically incompetent. > > > > This does not imply that D can be removed. > > > It does however imply that there is no single element > > of D that cannot be removed. That this does not > > imply that D can be removed is a result that > > you do not like, but it is not a contradiction. > > It is simple mathological blathering to insist that |N contains only > numbers that can be removed from |N but that not all natural numbers > can be removed from |N.
Nonsense, Removing any member of |N from |N leaves a proper subset of |N. However, removing FISONs from the set of all FISONs of |N may well leave enough (infinitely many) to have their union equal |N.
Being unable to understand this seems to be WM's personal pons asinorum. > > It is a contradiction with mathematics, namely with the fact that > every non-empty set of natural numbers has a smallest element. >
Another wild false claim by WM made, as usual, without proof. --