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SteveL27
SteveL27 is offline
#20
Feb6-12, 02:00 PM
P: 799
Quote Quote by Leucippus View Post

That would never happen in this process. That's obvious.
Leu, one point of confusion here is that you are thinking of Cantor's proof as a process. You are adding one row at a time and looking at the antigiagonal and then talking about rectangles.

All of that is erroneous thinking and has nothing to do with Cantor's proof.

Cantor asks us to consider any complete list of real numbers. Such a list is infinite, and we conceptualize it as a function that maps a number, such as 47, to the 47-th element on the list. There's a first element, a 2nd element, and DOT DOT DOT. We assume that ALL of these list entries exist, all at once.

Then we construct the antidiagonal. It's clear that the antidiagonal can't be on the list ... because it differs from the n-th item on the list in the n-th decimal place. (Or binary place if you do the proof in base-2).

Since we started by assuming we had a list of all reals; and we just showed that any such list must necessarily be missing a real; then it follows that there can be no such complete list in the first place.

I suggest that you make an attempt to fully understand this beautiful proof on its own terms. There is nothing here about processes and nothing here about rectangles. You are introducing those irrelevant concepts on your own and losing sight of the proof itself.