Could you check if my following attempt is O.K ?
Since every recursive relation is representable in the theory by assumption, there is a formula represents corresponding to in the theory (Enderton 2nd edition, pp 185).
Let be the Gödel number of .
Let be .
Then, says that no number is the value of (Enderton 2nd edition, pp 185) at a deduction of .
If , there is a contradiction explained in the textbook (Enderton 2nd edition, pp 185).
Therefore, we have .
Now, we need to show that .
Suppose to the contrary that .
By -consistency, it cannot be for all . It follows that it cannot be for every . Now, . It means is false in , contradicting that is true in (Enderton 2nd edition, pp 185).