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

 Messages: [ Previous | Next ]
 Robin Chapman Posts: 2,247 Registered: 12/6/04
Re: Cantor's diagonal argument.
Posted: Oct 3, 2001 10:03 AM

"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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Date Subject Author
10/3/01 Giles Redgrave
10/3/01 Jan Kristian Haugland
10/3/01 Robin Chapman
10/3/01 Clive Tooth
10/3/01 Christian Bau
10/3/01 briggs@encompasserve.org
10/3/01 Randy Poe
10/4/01 Giles Redgrave
10/5/01 Giles Redgrave
10/5/01 Jan Kristian Haugland
10/5/01 Dave Seaman
10/5/01 Christian Bau
10/5/01 Daryl McCullough
10/5/01 Steven Taschuk
10/5/01 Virgil
10/5/01 Tralfaz
10/5/01 Virgil
10/5/01 John Savard
10/12/01 Steve Brian
10/12/01 Virgil
10/4/01 Nico Benschop
10/3/01 Steven Taschuk
10/3/01 Andy Averill
10/3/01 Virgil
10/5/01 John Savard