Or is there some sort of proof that all spaces can be categorized into one of seven types of information spaces?

Hardly.. we haven't even touched things like modular spaces, yet.

Don't put too much weight on anything I say as a complete and accurate statement of mathematical truth. I'm chopping things up like mad, to keep from overloading everyday humans with the kinds of details necessary to be mathematical rigorous. At the same time, I'm trying not to stray too far from what a real mathemetician would be willing to accept in casual conversation.

Languages like Macsyma handle irrational numbers nicely.

Hmm.. interesting.

Personally, I'd call that an algebraic package rather than a numerical package per se, but that's really not a point I'd want to argue in any detail. You're right that the formula in question does represent a whole set of irrationals, which just goes to show that you can change the ground rules completely by changing your basic assumptions. ;-)