Goedel numbers are integers; how could anything that is enumerated by Goedel numbers not be enumerable? S_I, S, and R are all enumerable. The original paper says that R is the set of partial mu-recursive functions, which means computable functions; and the number of computable functions is enumerable.
You seem to be using ‘enumerable’ to mean ‘countable’. (Perhaps you’re confusing it with ‘denumerable’ which does mean countable.)
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?
You seem to be using ‘enumerable’ to mean ‘countable’. (Perhaps you’re confusing it with ‘denumerable’ which does mean countable.)
RichardKenneway means “recursively enumerable”.
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?
Sure, R is recursively enumerable, but S and S_I are not.