This approach leaves open the possibility that f(x) = BB(p+d) right?
No, because f is assumed to be computable from the start, which BB is not (otherwise it could be used as a subroutine in a program that solves the halting problem).
No, because f is assumed to be computable from the start, which BB is not (otherwise it could be used as a subroutine in a program that solves the halting problem).