Date: Mar 26, 2013 5:52 PM
Subject: Re: Matheology � 224
WM <firstname.lastname@example.org> wrote:
> On 26 Mrz., 22:07, Virgil <vir...@ligriv.com> wrote:
> > > > 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.
> > Mathematical Induction does not require use of natural numbers, but only
> > of a set which is as inductive as (is order-isomorphic to) the set of
> > natural numbers, and my form satisfies that requirement.
> Your mess is not induction and is obviously not a proof. A proof by
> induction runs as I said:
That WM said something is very nearly proof that it is false.
And induction requires only an inductive set, as I described above,
which need not be the set of positive naturals.