Date: Mar 26, 2013 4:56 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224

On 26 Mrz., 21:17, Virgil <vir...@ligriv.com> wrote:
> In article
> <2dc8b38d-3376-4a6c-89e4-ad4b059d8...@r1g2000yql.googlegroups.com>,
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 25 Mrz., 23:12, Virgil <vir...@ligriv.com> wrote:
>
> > > Lets see WM's statement of the inductive principle.
>
> > Let P(1)
> > and let P(x) ==> P(x+1)

>
> > Then P(n) at least for every natural number.
>
> > Proof: For P(2) follows from P(1), P(3) follows from P(2), and so on.
>
> > More is not required.
>
> If proof is not required, or even possible, in any system in which
> induction, or some equivalent, is not assumed.
>
> One acceptable form of induction is:
>
> There exists a set of objects, N,  and a zero object, 0,  such that
>    1. 0 is a member of  N.
>    2. Every member of N has a successor object in N.
>    3. 0 is not the successor object of any object in N.
>    4. If the successors of two objects in N are the same,
>       then the two original objects are the same.
>    5. If a set, S, contains 0 and the successor object of every
>       object in S, then S contains N as a subset.


That is a definition of a sequence, not a proof by induction. It is
not even a definition of the natural numbers, because even the ordered
set
N = (0, pi, pi^2, pi^3, ...)
obeys your five points.

Regards, WM