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