Math 280 Incompleteness 1. Definability, Representability and Recursion We will work with the language of arithmetic: 0_; S;_ +_ ; ·_; <_ . We will be sloppy and won't write dots. When we say \formula,"\satisfaction,"\proof,"everything will refer to this language. We will use the standard model of arithmetic N = (!; 0; S; +; ·; <). 1.1 Definition: (i) A bounded existential quantification is a quantification of the form (9z)(z < x ^ φ) which we will abbreviate by (9z < x)φ. (ii) A bounded universal quantification has the form (8z)(z < x ! φ) which we will abbreviate by (8z < x)φ. 1.2 Fact: The formulae (8z < x)φ $ :(9z < x):φ (9z < x)φ $ :(8z < x):φ are provable in predicate calculus. Proof: Exercise. 1.3 Definition: (i) A formula is bounded iff all definitions of this formula are bounded. We also say “Σ0"or “∆0"for bounded. (ii) A formula is Σn iff it has the form (9x1; :::; x1 )(8x2; :::; x2 )(9x3; :::; x3 ) ··· (Q(xn; :::; xn )) 1 l1 1 l2 1 l3 1 ln where is bounded. (iii) Πn formulae are defined dually. Here we start with a block of universal quantifiers. (iv) So Σn and Πn can be defined inductively: φ is Σn+1 iff φ has the form (9x1) ··· (9xl) where is Πn. Dually for Πn+1. 1.4 Definition: n A relation R(x1; :::; xn) ⊆ ( N) is Σl−definable iff R has a Σl−definition over N i.e. iff there is a Σl n formula φ(x1; :::; xl) such that for all a1; :::; an 2 ( N), R(a1; :::; an) iff N φ[a1; :::; an]: Similarly: R is Πl iff it has a Πl definition over N. 1 We also say that R is ∆l iff R is both Σl and Πl. 1.5 Definition: Robinson Arithmetic is the following set of axioms: (S1):S(x) = 0 (S2)S(x) = S(y) ! x = y (L1):x < 0 (L2)x < S(y) $ (x < y _ x = y) (L3)x < y _ x = y _ y < x (A1)x + 0 = x (A2)x + S(y) = S(x + y) (M1)x · 0 = 0 (M2)x · S(y) = x · y + x We denote these axioms by RA. We view these axioms as sentences, i.e. we frequently replace them by their universal closures. 1.6 Definition: Let Σ be a set of sentences. We say that a relation R ⊆ (nN) is representable in Σ iff there is a a1 a formula φ(x1; :::; xl) such that for all a1; :::; an 2 N : if R(a1; :::; an) then Σ ` φ(S (0); :::; S l (0)); if a1 a :R(a1; :::; an) then Σ ` :φ(S (0); :::; S l (0)): In this situation we say that φ represents R in Σ. Here Sa(0) is the term S(S(··· S(0) ··· )). Obviously (Sa(0))N = a. | {z } a n We say that a formula φ(x1; :::; xn) weakly represents R in Σ iff R = fha1; :::; ani 2 ( N)jΣ ` a1 an a1 an φ(S (0); :::; S (0))g. So in the above format: If R(a1; :::; an) then Σ ` φ(S (0); :::; S (0)). If a1 an :R(a1; :::an) then Σ 0 φ(S (0); :::; S (0)). n Given a function f :( N) ! N we say that a formula φ(x0; x1; :::; xn) represents/weakly represents f in Σ iff φ represents/weakly represents the relation R(y; x1; :::; xn) ≡ y = f(x1; :::; xn). n A formula φ(x1; :::; xn) is numeralwise determined in Σ iff for every a1; :::; an 2 ( N) either Σ ` φ(Sa1 (0); :::; San (0)) or else Σ ` :φ(Sa1 (0); :::; San (0)). 1.7 Remarks: Assume Σ ⊆Th(N), in particular this applies to RA. n (i) If φ(x1; :::; xn) is numeralwise determined, then φ represents a relation R ⊆ ( N). (This actually holds for any Σ.) (ii) If φ represents R then φ weakly represents R. (Because Σ is consistent.) This actually is true for any consistent Σ. n (iii) If φ weakly represents R in Σ then φ defines R in N. R = fha1; :::; ani 2 ( N)jN φ[a1; :::; an]g. 1.8 Definition: 2 (a) The following functions are called basic: c0 : N ! N defined by c0(x) = 0 S : N ! N defined by S(x) = x + 1 n n n For each 1 ≤ k ≤ n : Pk :( N) ! N defined by Pk (x1; :::; xn) = xk These functions are called projections. (b) We introduce the following operators on partial functions: n k - Composition: If we have functions g :( N) ! N and f1; :::; fn :( N) ! N then the composition k h is a function h :( N) ! N defined by h(x1; :::; xk) = g(f1(x1; :::; xk); :::; fn(x1; :::; xk)). - Primitive Recursion: Given functions f :(nN) ! N and g :(n+2N) ! N we define a function h :(n+1N) ! N by h(0; x1; :::; xn) = f(x1; :::; xn) h(y + 1; x1; :::; xn) = g(y; h(y; x1; :::; xn); x1; :::; xn): - µ−operator. Given a partial function g :(n+1N) ! N we define a partial function f :(nN) ! N by f(x1; :::; xn) =the least y such that (i) all values g(0; x1; :::; xn); :::; g(y; x1; :::; xn) are defined. 0 0 (ii) g(y ; x1; :::; xn) 6= 0 for all y < y. (iii) g(y; x1; :::; xn) = 0. 1.9 Remark: From the point of view of programming: - Composition corresponds to calling subroutines. - Primitive recursion corresponds to cycles. - µ−operator corresponds to search. 1.10 Definition: A partial function is recursive iff it can be finitely generated from basic functions using the three operators: composition, primitive recursion and the µ−operator. A partial function is primitive recursive iff it can be finitely generated from the basic functions using the operators composition and primitive recursion. For \primitive recursive, "we write briefly \p.i." Note: As all basic functions are total and the operators composition and primitive recursion preserve totality, all primitive recursive functions are total. 1.11 Main Theorem: (a) Let f be a partial function. TFAE: (i) f is Σ1−definable in N. (ii) f is weakly representable in RA. (iii) f is recursive. 1+n (b) For each n > 0 there is a partial recursive function Φn :( N) ! N which is universal for all partial recursive functions with n arguments. This means: if f :(nN) ! N is a partial recursive function then there is a number e 2 N such that for all a1; :::; an 2 N : f(a1; :::; an) ' Φn(e; a1; :::; an). 3 n 1+m (c) For 1 ≤ m < n there are primitive recursive functions sm :( N) ! N such that for every e; a1; :::; an 2 N : n Φn(e; a1; :::; an) ' Φn−m(sm(e; a1; :::; am); am+1; :::an): n Moreover, the functions sm are injective in each coordinate. (c) is called the s-m-n-theorem or the parameter theorem. 1.12 Definition: n Let Φn be the universal function from T.1.11. If f :( N) ! N is a partial recursive function, and e 2 N is such that f(x1; :::; xn) ' φn(e; x1; :::; xn) then e is called a G¨odelnumber of f. 1.13 Corollary: (a) The functions Φn are not total. (b) Any partial recursive function f :(nN) ! N has infinitely many G¨odelnumbers. Proof. Do this for n = 1 : (a) Since Φ1(x0; xn) is recursive, so is the function f(x0) ' Φ1(x0; x0) + 1. This is because f(x0) ' 1 Φ1(x0; p1(x0)) + 1. Now let e be a G¨odelnumber for f. So: f(x0) ' Φ1(e; x0) for all x0. That is, Φ1(x0; x0)+1 ' Φ(e; x0) for all x0, so in particular x0 = e. Therefore, Φ1(e; e)+1 ' Φ(e; e). (b) Given f : N ! N recursive, consider the function g : N × N ! N defined by g(x1; x2) ' f(x2). 2 This is again recursive, as g(x1; x2) ' f(p2(x1; x2)). Let e be a G¨odel number for g. So 2 f(x2) ' g(x1; x2) ' Φ2(e; x1; x2) ' Φ1(S1 (e; x1); x2): 2 2 So each S1 (e; x1) is a G¨odelnumber for f. But we also said that S1 is injective in each coordinate. 2 So the set fS1 (e; x1)jx1 2 Ng is an infinite set of G¨odelnumbers for f. 2. From Σ1−Definability to Representability 2.1 Lemma: For each n > 0: RA` x < Sn+1(0) $ x = 0 _ x = S(0) _···_ x = Sn(0). Proof. Induction on n: for n = 0, the axiom L2 says that x < S(y) $ x < y _ x = y. Since 0 is substitutable for y in this formula, RA` x < S(0) $ (x < 0 _ x = 0), using a lemma from predicate logic. But axiom L1 says :x < 0. Using propositional logic, we arrive at RA` x < S(0) $ x = 0. n ! n + 1: This time we substitute Sn+1(0) for y in L2, we get RA` x < Sn+2(0) $ (x < Sn+1(0) _ x = Sn+1(0)): By the IH: RA` x < Sn(0) $ (x = 0 _···_ x = Sn(0)). Using predicate logic, we can replace x < Sn+1(0) by (x = 0 _···_ x = Sn(0)). 2.2 Theorem: a1 an Let φ(x1; :::; xn) be a bounded formula and a1; :::; an 2 N. If N φ[a1; :::; an] then RA` φ(S (0); :::; S (0)) (∗). 4 Proof. Induction on complexity of φ. We will be showing that (∗) holds at φ and :φ at each step of induction. Claim 1: (i) If m = n, then RA` Sm(0) = Sn(0).