Date: Apr 4, 2013 4:30 PM
Subject: Re: Matheology � 224
WM <firstname.lastname@example.org> wrote:
> On 4 Apr., 19:45, William Hughes <wpihug...@gmail.com> wrote:
> > On Apr 4, 6:37 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 4 Apr., 18:13, William Hughes <wpihug...@gmail.com> wrote:
> > > > On Apr 4, 5:15 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > On 4 Apr., 16:01, William Hughes <wpihug...@gmail.com> wrote:
> > > > <snip>
> > > > > > If you remove "every finite line"
> > > > > > your are removing an infinite thing
> > > > > > "an infinite collection of finite things"
> > > > > If an infinite collection of infinite things exists actually, i.e., IF
> > > > > it is not only simple nonsense, to talk about an actually infinite set
> > > > > of finite numbers, then I can remove this infinite thing because it
> > > > > consists of only all finite things for which induction is valid.
> > > > Nope. The fact that the collection contains only things for which
> > > > induction is valid, does not mean induction is valid for the
> > > > collection.
> > > And you believe that, therefore, always elements must exists which in
> > > principle are subject to induction but in fact are not subjected to
> > > induction?
> > Nope, just that you can have a collection where everything in the
> > collection
> > is subject to induction, but where the collection itself is not
> > subject to
> > induction.
> If the collection is something else than all its elements, then you
> may be right.
A statement about the members of a set may not be true when referring to
the set itslef. A set of even integers is not itself an even integer.
WM has a long and disreputable history of being unable to distinguish
properties of a set from the properties of its members, or even between
the set itself and its members.
> Show this "else". In fact, an actually infinite set must
> constitute such a thing that cannot be removed when every finite set
> of elements is removed.
As soon as any member is removed, one no longer has the same set.
So an infinite set remains infinite when any one finite subset is
removed from it, but not when EVERY finite subset has been removed from
it as WM's claim implies.