I read that the set of partial recursive functions is recursively enumerable while the set of total recursive functions is not. Isn't the set of total recursive functions a proper subset of the set of partial recursive functions?
The argument for recursively enumerability of the set of partial recursive functions is that you can list the programs that compute them in a lexicographic order and use the index in that ordering as an argument of a function with two variables. The diagonal argument is used to show that the set of total recursive functions is not recursively enumerable. But why would the first argument fail in case of total rec. functions? Could anyone clarify this at all?