Date: Mar 25, 2013 7:26 AM
Subject: Re: Matheology § 224
On 24 Mrz., 22:35, Virgil <vir...@ligriv.com> wrote:
> The theorem does not cover what will transpire when two or more lines,
> along with all their predecessors, are removed.
There is no reason to remove more than one line with all its
predecessors, because it can be proved that all lines are predecessors
of a line, since there is no line without follower.
> So it is of some interest to note that for any set of lines having a
> maximal line in it,
Does induction not hold for the infinite set of naturals?
Or is there a maximal element in that set?
Or what else can be argued?
> > > 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.
> Except that no inductive argument will go from removing a finite set of
> lines to removing an infinite set of lines,
Induction holds for the infinite set of naturals and for the infinite
set of lines. Otherwise it would be superfluous.