Arithmetic is undecidable, in other words the set of Godel numbers of theorems of arithmetic is not recursive, and so there is no algorithm/ recursive function to decide if a statement is provable or not.
The Tarski undefinability theorem for arithmetic states that it is not possible to express by an arithmetic formula the set of Godel numbers of theorems of arithmetic. But arithmetic formulas contain the set of recursive functions, so this implies that the above set is not recursive.
Does all this make sense?