Re: Cantor's diagonal proof wrong?

From: Dave Seaman (dseaman_at_no.such.host)
Date: 11/21/04 

Date: Sun, 21 Nov 2004 22:19:04 +0000 (UTC)

On 21 Nov 2004 17:54:27 GMT, Curt Welch wrote:
> Dave Seaman <dseaman@no.such.host> wrote:

>> Do you agree that |N| < |P(N)|?

> For finite sets, it's obvious that the size of N is < size of the P(N).

N is not a finite set. N is the set of natural numbers. Cantor's proof
that |X| < |P(X)| makes no mention of whether X is finite or infinite,
nor does it need to.

> For infinite sets, the question seems to me to be invalid to ask.

Nonsense. The question being asked is whether you agree that

        1) there is an injection f: X -> P(X), and
        2) there is no surjection from X onto P(X).

You may possibly misunderstand the proof, but there is no doubt about
whether the question is meaningful. The question doesn't even mention
whether X is finite or infinite.

Notice that 1) means |X| <= |P(X)|, and 2) implies there is no bijection,
and therefore equality does not hold.

> It's exactly like the following word problem: If we built one machine to
> count to infinity which can count one number per second, and a second
> machine which can count at the speed 2^T numbers per second, where T is how
> long the machines have been running, which machine will finish first?

No, it's not even remotely like that problem. You didn't even mention
anything about injections or surjections.

> That question is just flat out invalid to ask. There is no answer.

Agreed. Now, can we get back to the question I originally asked?

> But, in set theory, it seems the question is not seen as invalid to ask, so
> I don't understand set theory yet. And if I don't understand set theory, I
> can't answer your questions about set theory.

Then perhaps the explanation above will help. 

-- 
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>

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