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?
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.
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? > > Once we remove one line, we are left with > a new set of unnecessary lines. We can > remove one of these lines. > From induction we get that > any finite set of lines can > be removed.
More. From induction we get that every set is finite. The error of matheology is to assume that there is more than every finite set, but this "more" obviously cannot be determined by induction (because it is not part of mathematics). Therefore any desired property could be attributed to this "more". However, it is not good taste to attribute properties like "unnatural" or "green" or so to this "more".
Our result is: If all natural numbers can be reached by induction, then nothing remains that makes the set |N actually infinite, because |N is not more than all natural numbers. And if not all natural numbers can be reached by induction, then there must be a first one of the "more" - at least, if the more consists of natural numbers.
You see, your position is untenable - at least in mathematics.