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Topic: Cantor's diagonal argument.
Replies: 24   Last Post: Oct 12, 2001 5:16 PM

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Robin Chapman

Posts: 2,247
Registered: 12/6/04
Re: Cantor's diagonal argument.
Posted: Oct 3, 2001 10:03 AM
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"Jan Kristian Haugland" <> wrote in message

> 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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