You're right! But I may still be right that the set of functions in R is enumerable. (Not that it matters to my post.)

There is a Turing function that can take a Goedel number, and produce the corresponding Goedel function. If you can define a programming language that is Turing-complete, and for which all possible strings are valid programs, then you just turn this function loose on the integers, and it enumerates the set of all possible Turing functions. Can this be done?