2. Let be a theory in a recursively numbered language, and assume that there is an interpretation of into . Show that is strongly undecidable; i.e., whenever is a theory in the language for which is consistent, then is not recursive.
This is my first attempt to the sketch of the proof based on Theorem 37D. Could you check the following?
Let be such an interpretation of into for which . Let be the consistent theory . Let be the corresponding theory in the language of number theory. Since is the consistent theory, is a consistent theory such that (from Section 2.7). By Theorem 35C, is not recursive, which follows that is not recursive (by the arguments in page 273).
We also have iff , where is the conjunction of the members of .
Therefore, is not recursive, nor is .