Date: Mar 26, 2013 4:56 PM
Subject: Re: Matheology § 224
On 26 Mrz., 21:17, Virgil <vir...@ligriv.com> wrote:
> In article
> 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
N = (0, pi, pi^2, pi^3, ...)
obeys your five points.