```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 isnot even a definition of the natural numbers, because even the orderedsetN = (0, pi, pi^2, pi^3, ...)obeys your five points.Regards, WM
```