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Math Forum » Discussions » sci.math.* » sci.math.independent

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

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Daryl McCullough

Posts: 502
Registered: 12/8/04
Re: Cantor's diagonal argument.
Posted: Oct 5, 2001 1:26 PM
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g.d.redgrave@elostirion.freeserve.co.uk (Giles) says...

>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.

No. It would be a correct proof by induction if
you replace 3. by "Therefore, for each n, A_n is
finite." That doesn't imply that every set of
naturals is finite, only those sets that happen
to equal A_n for some n.

--
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY

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