Re: paradoxes of randomness

The standard proof that all positive integers are interesting goes like this: - 1 is the smallest positive integer. That's interesting. - Suppose that you've proven that 1....N are interesting. Then either N+1 is interesting, and you continue the induction process, or - N+1 is the smallest integer that's not interesting. But that's interesting in itself - so N+1 is interesting. You can extend this to all integers, and to the rational numbers using the kinds of ordering that some of Cantor's proofs did. Doesn't work for real numbers, though - you can have a "nothing between X and Y is interesting, but X and Y are", without having any smallest number above X.