Use the recursion theorem to give an alternative proof of Rice’s theorem in Problem 5.28.

Problem 5.28.

Let P be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turingmachine’s language has property P is undecidable.

In more formal terms, let P be a language consisting of Turing machine descriptions where P fulfills two conditions. First, P is nontrivial—it contains some, but not all, TM descriptions. Second, P is a property of the TM’s language—whenever L(M₁) = L(M₂), we have 〈M₁〉 ∈ P iff 〈M₂〉 ∈ P. Here, M₁ and M₂ are any TMs. Prove that P is an undecidable language.