NiesBookWithErrata

1.1 The basic concepts 51.1.5 Recursion Theorem. Let g : N ↦→ N be computable. Then there is an esuch that Φ g(e) = Φ e . We say that e is a fixed point for g.Proof. By the Parameter Theorem 1.1.2 there is a computable function q suchthat Φ q(e) (x) ≃ Φ g(Φe(e))(x) for all e, x. Choose an i such that q = Φ i , thenSo e = Φ i (i) =q(i) is a fixed point.Φ q(i) = Φ Φi(i) = Φ g(Φi(i)). (1.2)We obtained the index i for q effectively from an index for g, by the uniformityof the Parameter Theorem. Thus, if g is a computable function of two arguments,we can compute a fixed point e = f(n) for each function g n given by g n (e) =g(e, n). Taking the uniformity one step further, note that an index for f can beobtained effectively from an index for g. This yields an extended version:1.1.6 Recursion Theorem with Parameters. Let g : N 2 ↦→ N be computable.Then there is a computable function f, which can be obtained effectivelyfrom g, such that Φ g(f(n),n) = Φ f(n) for each n.✷The incompleteness theorem of Gödel (1931) states that for each effectively axiomatizablesufficiently strong consistent theory T in the language of arithmetic one canfind a sentence ɛ which holds in N but is not provable in T . Peano arithmetic is anexample of such a theory. The incompleteness theorem relies on a fixed point lemmaproved in a way analogous to the proof of the Recursion Theorem. One represents aformula σ in the language of arithmetic by a natural number σ. This is the analog ofrepresenting a partial computable function Ψ by an index e, in the sense that Ψ = Φ e.Notice the “mixing of levels” that is taking place in both cases: a partial computablefunction of one argument is applied to a number, which can be viewed as an index fora function. A formula in one free variable is evaluated on a number, which may be acode for a further formula. The fixed point lemma says that, for each formula Γ(x) inone free variable, one can determine a sentence ɛ such thatT ⊢ ɛ ↔ Γ(ɛ).Informally, ɛ asserts that it satisfies Γ itself. Roughly speaking, if Γ(x) expresses thatthe sentence x is not provable from T , then ɛ asserts of itself that it is not provable,hence ɛ holds in N but T ⊬ ɛ.In the analogy between Gödel’s and Kleene’s fixed point theorems, Γ plays the roleof the function g. Equivalence of sentences under T corresponds to equality of partialcomputable functions. One obtains the fixed point ɛ as follows: the map F (σ) =σ(σ),where σ is a formula in one free variable, is computable, and hence can be representedin T by a formula ψ in two free variables (here one uses that T is sufficiently strong;we skip technical details). Hence there is a formula α expressing “Γ(F (σ))”, or moreprecisely ∃y[ψ(σ,y)&Γ(y)]. Thus, for each formula σT ⊢ α(σ) ↔ Γ(σ(σ)).Forming the sentence σ(σ) is the analog of evaluating Φ e(e), and α is the analog ofthe function q. Now let ɛ be α(α). Since α is the analog of the index i for q, ɛ (that is,the result of evaluating α on its own code number) is the analog of Φ i(i) (the resultof applying q to its own index). As in the last line (1.2) of the proof of the RecursionTheorem, one obtains that T ⊢ ɛ ↔ α(α) ↔ Γ(α(α)).✷