Re: Cantor's diagonal proof wrong?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 11/21/04 

Date: Sun, 21 Nov 2004 06:09:42 -0600

On 20 Nov 2004 21:52:40 GMT, curt@kcwc.com (Curt Welch) wrote:

>[...]
>
>What I actually said is that infinity is two separate things, not one.
>It's the algorithm which defines it, and it's the result of the algorithm
>as it runs.

By infinity here we mean infinite sets, right? No, infinity is not
defined by an algorithm.

>We can fully describe how to count with the natural numbers, but we can
>never actually use that procedure to count to infinity.

That's true. And totally irrelevant.

>When you manipulate the description for counting, you are working with one
>type of infinity. And there is no problem doing that.
>
>But, if you make an argument based on the idea that what you manipulate is
>the result of the counting procedure (i.e. all the natural numbers),
>instead of the procedure itself, you are talking nonsense.
>
>So, once again, there are two things that people confuse to be one when
>they talk about infinity. One thing is valid to work with (the algorithm),
>the other thing (all the output) is not.
>
>Every time I talk about the half that is invalid to work with, you are
>unable to see I've been talking about only one of the two halves, and you
>think I'm talking about the whole thing, and you think I have some problem
>with working with all types of infinity.
>
>You will never follow my logic until you understand I'm breaking the idea
>of infinity into two halves and making different statements about each
>half.

_nobody_ is able to follow your logic. This includes a lot of very
smart people who've thought a lot about all these things. But
the only possibility is that you're right and _everyone_ else is
wrong.

Amusingly, that's the only possibility even though the source of the
problem is what various statements _mean_!

************************

David C. Ullrich

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