Date: Apr 4, 2013 4:45 PM Author: mueckenh@rz.fh-augsburg.de 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

induction?

> 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

remaining.

>

> [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?

Regards, WM