For your final question, I find it hard to imagine a notion of “formal system” that does not just mean “language.”
How about http://en.wikipedia.org/wiki/First-order_logic#Syntax ?
So you agree that “provability oracle for an RE formal system” is the same as “membership oracle for an RE language” and your question is trivial?
ETA: No, first order languages does restrict the set of languages. But I object to this usage. “Formal systems” should include more general systems.
How about http://en.wikipedia.org/wiki/First-order_logic#Syntax ?
So you agree that “provability oracle for an RE formal system” is the same as “membership oracle for an RE language” and your question is trivial?
ETA: No, first order languages does restrict the set of languages. But I object to this usage. “Formal systems” should include more general systems.