In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 23 Mrz., 23:36, William Hughes <wpihug...@gmail.com> wrote: > > On Mar 23, 11:08 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 23 Mrz., 21:26, William hHughes <wpihug...@gmail.com> wrote: > > > You claim that no finite line of the set changes the union. > > > > There is no single finite line such that the removal of this one line > > changes the union. > > This holds for every line and all its predecessors, i.e., for the > whole potentially infinite set
So it only holds for actually finite sets. Since all of WM's potentilly infinite sets are actually finite sets, or perhaps foggily finite sets whose membership is ambiguuos. >
> > > You claim that when every finite line which does not change the union, > > > is deleted, then the union is changed.
When every member of a set is deleted, the union of it becomes empty. > > > > When every finite line with the property that when it alone is > > removed then the union is not changed, is deleted, then the union > > is changed. > > That is an unconfirmed statement.
It follows directly from "the union of the empty set is empty".
> And it is wrong, if every well- > defined set of natural numbers has to have a least element.
Unless WM can find a line whose removal will make the union of all remaining lines into a proper subset of the union of all lines, he is wrong.
> Do you > accept this theorem?
It isn't a theorem when provably false, and it is provably false.
> Do you agree that the definition "line of the list that does not > change the union of all lines" is well defined?
It is totally ambiguous. In any set of sets no one of the member sets has any effect on the union of the set of sets, unless it is removed or otherwise changed, which WM did not require or specify.
In ZF, one has the set of von Neumann naturals in which each "line" is also a natural, so that the set of naturals is the same as the set of lines.
And here the union of the set of all naturals is just the set of naturals.
And here the union of ANY infinite subset of the set of naturals is still the set of naturals.
But the union of any finite subset of the set of naturals is Not the set of naturals but is merely the maximal natural of that finite set.
Note that for any kind of set of naturals, at least when outside Wolkenmuekenheim, a subset is finite if and only if it has a maximal member and a set is infinite if and only if it does not have a maximal member. --