[moyix]

This reminds me of the self-defeating property of an "uninteresting" number -- a reasonable definition might be "any number that does not have any property of human interest", but then of course there is a smallest such number, and so that has the interesting property of being the first uninteresting number, a contradiction!

You're right though that a more precise definition of "real-world numbers" is needed, but I confess that my attempts to think of one in the past few minutes have been essentially circular (coming down to "the ones we know how to compute")!

[jfarmer]

Well, we can and have made the idea of a computable number precise: http://en.wikipedia.org/wiki/Computable_number

It's not clear whether the universe is computable, however, in the sense that we only find computable numbers in nature. This is kind of an epistemological catch-22, though. How would we know whether this were the case or not?