Thanks!
A^B is the set of functions from B to A. So 2^N is powerset of N (a function f from N to {0, 1} says, for each element of N, whether or not that element is in the subset defined by f), which is isomorphic to the reals. Perhaps this should go somewhere in one or more of the versions. I don't know any connection between this and P=NP (although I suppose it could be behind the exponential bounds on various things).