Several questions on MSE in recent months and most recently this one have made me feel that recursion theory is suffering from terminology bloat. Why have so many synonyms for “recursive” and “recursively enumerable” been introduced?
I can just about see why people might want to use “computable” for recursive or use “computably enumerable”, “Turing-acceptable” or “Turing-recognizable” for recursively enumerable.
I can’t understand why anyone would want to use “decidable” for recursive, “partially decidable” or “semidecidable” for r.e. unless they were talking about sets that are interpreted as logical judgments (like the word problem for group theory) rather than any other sets (like codes of Turing machines). So I think “decidable’ should be reserved for instances of some kind of Entscheidungsproblem.
I can just about see how someone could argue that “recognizable” intuitively suggests r.e. but not recursive. But I can’t really see any advantage in changing to use that terminology.
What is going on with the terminology of recursion theory?