In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 24 Mrz., 22:35, Virgil <vir...@ligriv.com> wrote: > > > > > The theorem does not cover what will transpire when two or more > > lines, along with all their predecessors, are removed. > > There is no reason to remove more than one line with all its > predecessors, because it can be proved that all lines are > predecessors of a line, since there is no line without follower. > > > > So it is of some interest to note that for any set of lines having > > a maximal line in it, what WM claims (that one can remove any line > > frm any set of lines without affecting the union of the set of lines) > > is false. At least everywhere outside Wolkenmuekenheim
> > Does induction not hold for the infinite set of naturals?
Not in Wolkenmuekenheim.
> > > > > > > > > > > > > > > > > > > > However, we do not know what will happen if we remove an > > > > infinite number of finite lines. > > > > > That's why we use induction. > > > > Except that no inductive argument will go from removing a finite > > set of lines to removing an infinite set of lines, > > Induction holds
It does not hold that removing the largest member of a set having a unique largest member reproduces the original set exactly.
At least it does not do so outside of Wolkenmuekenheim, even though WM claims ti does so everywhere. --