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.
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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 .