Re: Cantor and the binary tree

In article <MPG.1d0da54875d1ec5e989691@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> In article <ITSnetNOTcom#virgil-4535B2.22320405062005
> @comcast.dca.giganews.com>, ITSnetNOTcom#virgil@xxxxxxxxxxx says...
> > In article <MPG.1d0d811d66d993d9989687@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >
> > > In article <8t-dnQXYYPWvED3fRVn-uQ@xxxxxxxxxxx>, nowhere@xxxxxxxxxxx
> > > says...
> > > > W. Mueckenheim wrote:
> > > > > "If n e N then n+1 e N" describes the action to consider n, to find
> > > > > out whether it is a natural, perhaps by considering n-1 ..., and,
> > > > > finally, to calculte n+1.
> > > >
> > > > In induction proofs, the n is a variable, so one only has to prove for
> > > > arbitrary n (variable) if n is in the set then so is n + 1. An infinite
> > > > number of operations are NOT necessary. Which is the entire point of
> > > > the
> > > > Induction Axiom. One can reduce an infinite set of implications to just
> > > > two assertions.
> > > >
> > > > 1. 0 is in the the set
> > > >
> > > > 2. For arbitrary n is n in the set then n + 1 in the set.
> > > >
> > > > Two does the work of infinity.
> > > >
> > > > If you cannot grasp how variable (arbitrary) quantities function in a
> > > > prood, you really should give up mathematics and take up the healthy
> > > > sport of curling.
> > > >
> > > > Bob Kolker
> > > >
> > > If you want to claim you have proven something for all members of an
> > > infinite set using induction, you better make sure your property is true
> > > at n=oo, or you haven't proven it for all members.
> >
> > Where does it say anything like that in any statement of the inductive
> > axiom?
> It claims to prove a fact for every member of the entire infinite set of
> natural numbers. That means each and every one, so if your proof doesn't
> hold for an infinite number of iterations, it doesn't prove it for the
> entire set, and is therefore a missapplication of induction.

You miss the point of induction. True for 1 and truth inherited by each
successor means truth for 1 and all its successors. Since these are alll
of them,it is then true for all of them. Infinitely many separate proofs
are not needed.
> >
> > In the absence of any such statement in any version of that axiom, it is
> > not a requirement!
> It is a logical consequence of the definition and the assumption of an
> infinite set of naturals.

We do not assume any such thing.

It is a consequence of the Cantor definition of finiteness of a set that
the set of naturals is provably not finite.
.

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