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Re: Cantor's diagonal argument.
Posted:
Oct 5, 2001 3:25 PM
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"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.
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