```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 andwill be applied after (finished infinite) sets will have beenexorcized.> 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 ofyour phantasy sets like 1, 1/2, 1/3, ... is m(m+1)/2? This is provableusing 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 hisoriginal paper writes explicitly: "The sign a + 1 means *the successorof 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 matheology1/3 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 dervollständigen Induktion (oder des Schlusses von n auf n + 1) bekannteBeweisart wirklich beweiskräftig, und daß auch die Definition durchInduktion (oder Rekursion) bestimmt und widerspruchsfrei ist. DieseSchrift kann jeder verstehen, welcher das besitzt, was man dengesunden Menschenverstand nennt; philosophische oder mathematischeSchulkenntnisse sind dazu nicht im geringsten erforderlich.The conclusion from n on n + 1 is well known as complete induction (orrecursion).Everybody, who owns some common sense, can understand that, Dedekindwrites, - of course matheologians are excluded.Regards, WM
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