> Approximating a real number to an arbitrarily precise rational number can then be done by applying some natural number. But the given representation doesn't

> Approximating a real number to an arbitrarily precise rational number can then be done by applying some natural number. But the given representation doesn't allow you to determine what natural number n is needed to yield a desired approximation. Allowing that gives you the class of computable reals. For instance, one can require that f_x(n) be within 2^-n from x. The given representation can represent a larger class than just the computable reals. For example, it can represent Chaitin's Omega, the halting probability of a universal machine, since f_Omega(n) can just be the contribution from programs of size at most n that halt within n steps. Omega is only approximable from below, making it an enumerable rather than computable number. The latter are approximable both from below (with f_x(n+1) >= f_x(n)) and from above (with f_x(n+1) <= f_x(n)).