Date: Apr 4, 2013 4:45 PM
Subject: Re: Matheology § 224
On 4 Apr., 20:57, William Hughes <wpihug...@gmail.com> wrote:
> On Apr 4, 8:30 pm, WM <mueck...@rz.fh-augsburg.de> 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.
> No, a collection is no more and no less than "all its elements".
But an inductive set contains elements that are not subject to
> Note the "no less". A collection need not share a property
> that every one of its elements has. In this case
> every one of the elements of the collection has the property
> that it can be removed without changing the union.
> The collection does not have this property.
That is impossible if all elements can be removed.
Compare the collection of three elements. If they are gone, the
collection is empty - you may claim that then there is something
> [This holds for the "collection of all finite lines"
> other collections, e.g. a "collection of three lines"
> do have the property that they can be removed without
> changing the union]-
Why do you think that there is a difference?
Why do you not think (at least you have not yet mentioned it) that
Cantor's argument also cannot exhaust the complete set?