A device computes the Halting problem - it is able to return h(T, x) for any given Turing machine (computer program) T and input x, h(T, x) = 1 if and only if T halts on input x, and 0 otherwise. Computer scientists eagerly designed a hypercomputer based on this technology, and incorporated it within a standard universal computer. A program on this hypercomputer can, at each time-step, either (i) operate as a standard Turing machine, or (ii) instead of having the head moving left or right, it can choose to call upon a device to solve for h(T, x), where T, x are stored onto the tape in some standard format (e.g. by having one binary string representing T, followed by # and a second binary string representing x.) (a) Let P be the set of all programs that can be written on this hypercomputer. Argue why just as programs written on standard Turing machines, each element of P can be mapped to a unique natural number.