paradoxes of randomness

[Bill Stewart]

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.