Re: A question about Caontor's proof of the uncountability of the reals

On Mon, 27 Feb 2006 06:06:23 GMT, Michael Olea wrote:

This, I think, misconstrues my post. I have never had any trouble
understanding Cantor's diagonalization proof: assume the reals are
countable, show that this assumption leads to a contradiction, conclude
that therefore that the assumption was false. This all makes perfect sense.
It is only when the law of the excluded middle comes under fire - something
I've only given any attention recently - that there may be room for seeds
of doubt. Are there conditions under which (P) and (not P) are not the only
possibilities, and therefore establishing that not P is false need not
imply the P is true? So the real question was not about Cantor's proof of
the uncountability of the reals per se, but about the validity of proof by
contradiction in axiomatic systems where Godel's incompletenes theorems
apply.

Are you aware that direct proofs exist?

Let f: N -> R be an arbitrary mapping. Show that there exists x in R
that is not in the range of f, and hence f is not a surjection. Since f
is arbitrary, we conclude that no surjection exists.

This is a direct proof. There is no contradiction in sight, and no
reliance on excluded middle. Does this answer your objection?


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

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