Re: Cantor's diagonal proof wrong?
From: Todd Trimble (trimble1_at_optonline.net)
Date: 11/15/04
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Date: Mon, 15 Nov 2004 02:27:09 +0000 (UTC)
On 14 Nov 2004, Curt Welch wrote:
>Let me demonstrate.
>
>I claim that there is only one type of infinity. That there are just as
>many integers as there real numbers. (or more accurately, that the concept
>of the size of an infinite set is a contradiction in itself).
>
So you claim that all infinite sets have the same cardinality?
Given a bijection f: X --> P(X) between X and its power set P(X),
what do you say about A = {x in X: x not in f(x)}? Since f is a
bijection, there exists y in X such that f(y) = A. Does y belong
to A? If so, then y is not in f(y) [cf. defn. of A], i.e., y is
not in A. Does y not belong to A? If that's the case, then it
is false that y is not in f(y); therefore, it's true that y is
in f(y), and thus y is in A. Either way we reach a contradiction;
therefore there is no bijection between X and P(X).
This is a version of the diagonal argument. Please point out
why it's "wrong".
Todd Trimble
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