Date: Mar 24, 2013 11:03 AM
Subject: Re: Matheology § 224

On 24 Mrz., 14:36, William Hughes <> wrote:
> On Mar 24, 12:22 pm, WM <> wrote:

> > Induction holds for all elements of the
> > infinite set of natural numbers.

> And every one of these elements is finite.

Of course. That't why they all can be deleted from the list without
changing the infinite set |N in the list.
> Induction works for an infinite number of different
> things.
> Each of these things is finite.
> Induction does not work on something that is
> not finite (like an infinite number of finite numbers)

Induction works *for* an infinite number of naturals, namely on the
infinitely many elements of the set |N, but not *on* an infinite
number of naturals. Try to escape by prepositions? Now you get silly.
Take it as follows: Induction proves that every and all finite lines
of our list can be removed without changing the contents, the union of
the list. Otherwise you could name the first finite line that is not
subject to induction, which you cannot. There is no finite line that
must remain. Therefore the collection of removable lines is infinite.

Regards, WM