Re: Cantor and the binary tree
- From: Ron Sperber <ronsperber@xxxxxxxxxxxxx>
- Date: Tue, 24 May 2005 15:26:30 -0400
Robin Chapman wrote:
Robert Kolker wrote:
mueckenh@xxxxxxxxxxxxxxxxx wrote:
of paths always equals that of the nodes + 1. It is simply impossible to assume that one of these numbers becomes uncountably infinite while the other remains countably infinite.
"becomes"? Muck's fuzzy metaphors are sabotaging him again. The fact is that the nodes in this tree form a countable set and the paths form an uncountable set. "becoming" has nowt to do with that.
Wrong. 2^(aleph_0) > aleph_0.
List all the infinite binary sequences with a bijection to the integers. Now flip the n-th digit of the n-th sequence in the list. This cannot occur anywhere in the list. Contradiction. Such a bijection to the integers does not exist.
One can hardly imagine a simpler mathematical proof. Alas, it's still beyond the limits of Muck's feeble intellect :-(
It simply boggles my mind that this simple proof gives so many people such fits that they refuse to accept it. I continue to be sadly shocked by the number of posts on sci.math daily refuting Cantor's proof. Of course they are always fuzzy on details, but that's to be expected since they can't actually disprove it.
.
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