On Mar 21, 12:32 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 21 Mrz., 11:36, William Hughes <wpihug...@gmail.com> wrote: > > > On Mar 21, 11:21 am, WM <mueck...@rz.fh-augsburg.de> wrote:> On 21 Mrz., 08:57, William Hughes <wpihug...@gmail.com> wrote: > > > <snip> > > > > > 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? > > > You can only use induction to prove > > stuff about finite sets. > > In fact? That's amazing. So we cannot prove that all lines of the > infinite set of lines are unnecessary? >
We can prove that something is true for every member of an infinite set. We cannot prove that something is true for the set itself unless the set is finite.
> Note: For every finite set of natural numbers, we can look at all > elements, at least in principle. We do not need induction for fixed > finite sets. Induction is not *necessary* then, so to speak.
If you want to say that something is true for some particular finite set you do not need induction If you want to say that something is true for each finite set, then you do need induction. If you want to say that something is true for each infinite set then induction will not help.
> > I hope you see that your claim is nonsense, iff there is an infinite > set of natural numbers. > > > > > > 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. > > > Nope, their property of being unnecessary means > > that *any one* line can be removed. > > But you think that after all finite and unnecessary lines another one > is lurking like a dragon?
Now I think that after any finite set of unnecessary lines has been removed, there still remains an unnecessary line.