On 4/25/07, Nadav Har'El <[EMAIL PROTECTED]> wrote:
Uri, this is becoming (or was always) extremely off-topic.
Well, it is off topic.
Georg Cantor's beautiful proof that there are more real numbers than natural numbers (involving the diagonal of the real number's list) is one of the most striking - and self-evident - proofs that I've ever seen (my father showed it to me when I was a kid). Wikipedia (which I'm glad you're using as a source - I thought you thought it was the devil :-)) also has an article about this: http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
I read this article before writing you. Look what it says about the constructivist approach. Whether or not there are "more" real numbers than rational numbers, depends on how you define "more". Every language contains ambiguities. Proving that there are things which can't be expressed by any specific language is in itself inconsistent. For example, is "the smallest natural number that can't be expressed in less than 20 words" a number? Is there no such number? Anyway, proving that a specific problem is "hard" for any algorithm means something like "we can express more problems than we can express solutions in the language we are using". I'm arguing that if a function is proved to be not computable, then it's not well defined. The definition is not completely deterministic. I changed my mind about Wikipedia. It's a very good source for information on many subjects, and I use it a lot (especially the English version of it). But there are some cases where it is controversial, and I hardly ever write there at all. If I write anything, in most cases it will be deleted. So it is both good and bad, both one and zero, both god and the devil at the same time. I'm allowing myself to be inconsistent. Uri. ================================================================= To unsubscribe, send mail to [EMAIL PROTECTED] with the word "unsubscribe" in the message body, e.g., run the command echo unsubscribe | mail [EMAIL PROTECTED]
