Date: Mar 26, 2013 4:17 PM
Subject: Re: Matheology � 224
WM <firstname.lastname@example.org> 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.
So WM gets a failing grade!
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.