Date: Apr 3, 2013 3:22 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224

On 3 Apr., 20:25, Virgil <vir...@ligriv.com> wrote:
> In article
> <3d408b46-a74c-4a88-98ff-43d981d43...@t5g2000vbm.googlegroups.com>,
>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 3 Apr., 09:26, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <2569eb91-7037-483e-be2c-17fce8394...@j9g2000vbz.googlegroups.com>,

>
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > On 3 Apr., 00:29, Virgil <vir...@ligriv.com> wrote:
>
> > > > > The point being that removing one object from an infinite set does not
> > > > > diminish the infinite number left in the set

>
> > > > That is a good point. Alas induction holds for every natural number.
>
> > > No!
>
> > Your no is wrong. Induction holds for every natural number.
>
> It does not hold for any natural number as an individual other than when
> in the set of naturals or in some other inductive set.


It does only hold for the set because it holds for every individual.
Induction has been applied before there were sets in mathematics and
will be applied after (finished infinite) sets will have been
exorcized.

> It is the
> inductiveness of the set, not that it is necessarily a set of natural
> numbers in particular, that justifies induction.


Can you prove by induction that the sum of the first m elements of
your phantasy sets like 1, 1/2, 1/3, ... is m(m+1)/2? This is provable
using induction.

> For purposes of induction, one does not need natural numbers, though
> they can be used.


A gang of matheologians have distorted Peano's writings. Peano in his
original paper writes explicitly: "The sign a + 1 means *the successor
of a*, or *a plus 1*." And later he adds: "2 = 1 + 1, 3 = 2 + 1".
(Heijenoort p. 94)

Any idea why? Perhaps in your matheology 1/3 = 1/2 + 1? In matheology
1/3 is greater than 1/2.
>
>
> That the natural numbers form one example of an inductive set does not
> mean that that set is the only example.


Dedekind writes: den Nachweis, daß die unter dem Namen der
vollständigen Induktion (oder des Schlusses von n auf n + 1) bekannte
Beweisart wirklich beweiskräftig, und daß auch die Definition durch
Induktion (oder Rekursion) bestimmt und widerspruchsfrei ist. Diese
Schrift kann jeder verstehen, welcher das besitzt, was man den
gesunden Menschenverstand nennt; philosophische oder mathematische
Schulkenntnisse sind dazu nicht im geringsten erforderlich.

The conclusion from n on n + 1 is well known as complete induction (or
recursion).
Everybody, who owns some common sense, can understand that, Dedekind
writes, - of course matheologians are excluded.

Regards, WM