Date: Mar 24, 2013 7:22 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224
On 24 Mrz., 11:19, William Hughes <wpihug...@gmail.com> wrote:

> On Mar 24, 11:04 am, WM <mueck...@rz.fh-augsburg.de> wrote:

>

> Your proof covers all lines. We have for all lines l

> of the list.

>

> if l and all its predecessors are removed

> and no other line is removed,

> then the union of all lines is not changed"

>

> However, there is no information about what will

> happen if you try to apply this to two

> lines e.g. l along with all its predecessors

> and m along with all its predecessors.

Nothing will "happen". Induction holds for every line, so it holds for

all lines of the set of finite lines.

>

> Now it is easy to see what will happen in this

> case. Since we can replace l and m with

> one of either l or m, we know what will happen

> if we remove two lines.

>

> Since we can replace l,m and p with

> one of either l or m or p, we know what will happen

> if we remove three lines.

This way is not necessary, since the proof holds for every line

including all its predecessors. Therefore it holds for l, m, and p

(because they belong to a set of predecessors. This is a property of

the natural numbers and dos not require any further attention.

>

> It is easy to see we know what

> will happen if we remove a natural

> number of finite lines.

>

> However, we do not know what will happen

> if we remove an infinite number of

> finite lines.

That's why we use induction. Induction holds for all elements of the

infinite set of natural numbers.

Regards, WM