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

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 briggs@encompasserve.org Posts: 404 Registered: 12/6/04
Re: Cantor's diagonal argument.
Posted: Oct 3, 2001 10:59 AM

In article <c645a590.0110030525.199817f7@posting.google.com>, g.d.redgrave@elostirion.freeserve.co.uk (Giles Redgrave) writes:
> 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.
>
> I don't understand why you can't apply CDA to the natural numbers
> themselves. If we list the natural numbers padding to the left with
> zeros like so:
>
> ...000
> ...001
> ...002
> ...003
> .
> .
> .
>
> and apply CDA by adding one to the nth digit (from the right) of n and
> constructing our new number from these digits.
>
> 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.

Classic issue. The resolution is to realize that this number is
not a natural number. All natural numbers have finitely many
decimal digits. This number has infinitely many.

Getting a handle on the concept of "arbitrarily large but finite"
is sometimes a key to wrapping yourself around this conundrum.

The decimal expansion of any particular natural number might be
arbitrarily large. But it will always be finite.

This is different from the decimal expansions of real numbers. Some
(most) of those are infinite.

John Briggs

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