Date: Apr 4, 2013 5:28 PM
Author: Virgil
Subject: Re: Matheology � 224

In article 
WM <> wrote:

> On 4 Apr., 22:30, Virgil <> wrote:

> > A statement about the members of a set may not be true when referring to
> > the set itslef. A set of even integers is not itself an even integer.

> True. But if all even integers are removed, then nothing remains.

The empty set, which is a thing, remains.

> Induction is valid for the set of all even integers.

Standard Induction requires only a set which is order isomoprphic to the
set of naturals with the natural ordering on it

> Therefore it is
> possible to remove all of them, if it is pissble to remove 2 and with
> n also n + 2.

But what is the point of "removing all of them"?
> > So an infinite set remains infinite when any one finite subset is
> > removed from it, but not when EVERY finite subset has been removed from
> > it as WM's claim implies.

> If the set |N is actually infinite, then every finite subset of lines
> of the set
> 1
> 1, 2
> 1, 2, 3
> ...
> can be removed without changing the infinite union of all finite
> lines.

Not in English.

While removing ANY one finite such set can leaves an inductive set,
removing EVERY such set removes the union of all such sets and leaves
only the empty set, which is not inductive.