Our first main theorem in this chapter establishes the ‘recursive unsolvability of the self-halting problem’ for Turing machines. This is one of those pivotal results like the Diagonalization Lemma which at first sight can seem just an oddity but which entails a whole raft of important results. We use this theorem to establish (or re-establish) various claims about incompleteness and decidability.

We also prove a version of Kleene's Normal Form Theorem: this leads to yet another proof of incompleteness.

Two simple results about Turing programs

(a) As a preliminary, let's note

Proof sketch Use some system for Gödel-numbering sets of i-quadruples. For example, use powers of primes to code up single i-quadruples; then form the super g.n. of a sequence of codes-for-quadruples by using powers of primes again.

Now run through numbers e = 0, 1, 2, …. For each e, take prime factors of e, then prime factors of their exponents. If this reveals that e is the super g.n. of a set of i-quadruples, then check that it is a consistent set and hence a Turing program (that is an effective procedure, and the search is bounded by the size of the set of i-quadruples). If e is the super g.n. of some Turing program Π, put Πe = Π; otherwise put Πe = Π* (where Π* is some default favourite program). Then Π0, Π1, Π2, … is an effectively generated list of all possible Turing programs (with many repetitions).