Hm, I don't know if this is the step in the same direction, but the first thing that comes to mind for 2) is, if A were the set of natural numbers, then each finite subse can be encoded by a single number. E.g., the list of prime numbers is , , , , ..., so would be encoded as . By Fundamental Theorem of Arithmetic, factorization into primes is unique, so one gets the original set unambiguously from the code.