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Re: Cantor's diagonal argument.
Posted:
Oct 3, 2001 10:03 AM
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"Jan Kristian Haugland" <jkhaug00@stud.hia.no> wrote in message
news://Pine.GSO.4.05.10110031531300.2789-100000@svale.hia.no...
>
> On 3 Oct 2001, Giles Redgrave wrote:
>
> > I'm having a problem understanding Cantor's diagonal argument (CDA).
> > Specifically it's use in proving the uncountability of the reals from
> > 0 to 1.
> (...)
> > We then have a number (consisting of an infinite series of ones) that
> > is not in the original list because it is different from each n in
> > it's nth digit.
> >
> > But it is clear that this number *is* in the list because it is a
> > natural number.
>
> No, it is clear that it is _not_ a natural
> number, because it is not finite.
Agreed, but I always think it's unwise to mention finiteness
here since some people then drag the discussion into a
pseudo-philosophical quagmire. It is best to note, that
if a natural number has decimal digits a_i, then a_i = 0 for
all i > some N which we may take to equal a if we like.
Proof: induction. Then it's apparent that the "number" with
a_i = 1 for all i is no natural number.
Robin Chapman
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