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

 Messages: [ Previous | Next ]
 Steven Taschuk Posts: 134 Registered: 12/6/04
Re: Cantor's diagonal argument.
Posted: Oct 5, 2001 3:25 PM

"Giles Redgrave" <g.d.redgrave@elostirion.freeserve.co.uk> wrote:
> dseaman@seaman.cc.purdue.edu (Dave Seaman) wrote
> in message news:<9php6j\$bkl@seaman.cc.purdue.edu>...

...
> > No, the correct argument is
> >
> > 1. 1 is finite.
> > 2. For each n, if n is finite, then n+1 is finite.
> > 3. Therefore, each natural number is finite.
> >
> > The conclusion is a statement about each natural number, not about the
> > set of all natural numbers. That is how induction works.

>
> Why can't you apply the same inductive argument to sets of natural
> numbers.
>
> 1. A_1 has a finite number of elements
> 2. For each n, if A_n has a finite number of elements, then A_{n+1}
> has a finite number of elements.
> 3. Therefore, each set of natural numbers has finite size.
>
> This seems to me to be exactly the same argument.

...

Your statement (3) is too broad; the induction proof suggested by (1) and
(2) supports the conclusion:
For all n, A_n is finite.
This is not the same as saying that each set of natural numbers has finite
size; only those particular sets A_1, A_2, etc., are addressed. There are
lots of sets of natural numbers which are not among these; the set {2,3,5},
for example. More to the point at hand, it's easy to see that none of the
A_n is the set of all natural numbers, since A_n doesn't contain n+1.

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