Re: Orlow cardinality question

In article <MPG.1d20c7f4f967eb0989e48@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> *** T. Winter said:

> > > It does as n approaches N (aleph_0). If aleph_0 is the size of
> > > the set, then it is also the maximal element.
> >
> > Sorry, but you are guilty of circular reasoning. Aleph_0 is the
> > size of the set, but not a number according to Peano's axioms. The
> > induction axiom is about numbers only, so the axiom does not apply.
>
> I don't care what you think is in the set or out of it, but if
> induction proves anything then the size of the set of naturals is
> also the value of its maximal element.

Wrong! For every element in the set of naturals there is a larger
(later) element, its successor, which is different from it. So that a
largest or last is impossible.
.

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