```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 forall 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 lineincluding all its predecessors. Therefore it holds for l, m, and p(because they belong to a set of predecessors. This is a property ofthe 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 theinfinite set of natural numbers.Regards, WM
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