Undecidability and Recursive Inseparability

§1. Some Preliminary Theorems. we continue to let S be an arbitrary system, P be the set of Gödel numbers of the provable formulas of S and R be the set of Gödel numbers of the refutable formulas of S. Theorem 1. The set P̃* is not representable in S. Proof. This is the diagonal argument all over again. If H(v1) represents P̃* and h is the Gödel number of H(v1), the H[h̅] is provable in S iff h Ï p* iff d(h) ÏP iff H[h̅] is not provable in S, which is a contradiction. Theorem 1.1. If S is consistent, then P* is not definable in S. Proof. Suppose P* is definable in S. If S were consistent, then P* would be completely representable in S (cf. §3.1, Ch. 0). Hence P̃* would be representable in S, contrary to Theorem 1. Therefore, if S is consistent, then P* is not definable in S. Theorem 1.2. If the diagonal function d(x) is strongly definable in S and S is consistent, then P is not definable in S. Proof. Suppose d(x) is strongly definable in S. Since P* = d -1(P), then if P were definable in S, P* would be definable in S (by Th. 11.2, Ch. 0). Hence S would be inconsistent by Theorem 1.1. Exercise 1. Show that if S is consistent, then R* is not definable in S. Exercise 2. Show that if S is consistent, then no superset of R* disjoint from P* is definable in S, and no superset of P* disjoint from R* is definable in S. Exercise 3. Prove that if S is consistent and if the diagonal function is strongly definable in S, then no superset of P disjoint from R is definable in S. [This is stronger than Theorem 1.2.] §2. Undecidable Systems. A system S is said to be decidable (or to admit of a decision procedure) if the set P of Gödel numbers of the provable formulas of S is a recursive set. It is undecidable if P is not recursive. This meaning of ‘undecidable’ should not be confused with the meaning of ‘undecidable’ when applied to a particular formula (as being undecidable in a given system S).

Indexing

For the remaining chapters, we will need two basic theorems in recursive function theory—the enumeration theorem of Kleene and Post and the iteration theorem of Kleene. §1. Indexing. we wish to arrange all r.e. sets in an infinite sequence ω0, ω1, . . . ,ωn , . . . (allowing repetitions) in such a way that the relation xÎ ωy is r.e. we shall take the system (Q) as our basic formalism for recursive function theory. we know that (Q) is axiomatizable and that the representable sets of (Q) are precisely the r.e. sets. we define ωi - as the set of all numbers n such that Ei[ n̅ ] is provable in (Q). Equivalently, wi- is the set of all n such that r(i,n) Î P, where r(i,n) is the Gödel number of Ei[ n̅ ] and P is the set of Gödel numbers of the provable formulas of (Q). Since r(x,y) is a recursive function and P is an r.e. set, then the relation r(x,y) Î P is r.e., and this is the relation y Î ωx. Also, every r.e. set A is represented in (Q) by some formula Ei(v1); hence A = ωi. Thus every r.e. set appears in our enumeration. we call i an index of an r.e. set A if A = ωi. we let U(x,y) be the relation x Î ωy , and we see that this relation is r.e. Indexing of r.e. Relations. For each n ³ 2, we also wish to arrange all r.e. relations of degree n an in infinite sequence . . . Ron,R1n , . . . , Rnn , . . . in such a manner that the relation Ryn(x1, . . . ,xn) is an r.e. relation among x1, . . . , xn and y. To this end, it will be convenient to use the indexing of r.e. sets that we already have and to use the recursive pairing function J(x,y) and its associated functions Jn(x1,. . . ,xn) (cf. §4, Chapter 1).

Productivity and Double Productivity

Now we will give the promised applications of the (strong) recursion and double recursion theorems to the theory of productivity and effective inseparability (and also to double productivity—a double analogue of productivity—which we will define). §1. Weak Productivity. We recall that a set α is said to be co-productive under a recursive function g(x) if for every number i, such that ωi is disjoint from α , the number g(i) is outside both a and ωi. This, of course, implies the folloωing weaker condition: C1: For every i, such that ωi is disjoint from α and such that ωi contains at most one element, the number g(i) is outside both α and ωi

Symmetric and Double Recursion Theorems

We have proved that every completely E.I. pair of r.e. sets is D.U. In the next chapter we will show the stronger result that every E.I. pair of r.e. sets is D.U. The proof of this is based on the double recursion theorem of this chapter. Our original formulation of the double recursion theorem (T.F.S.) required the recursive pairing function J(x,y), not only for its proof, but for its very statement. In this chapter, we give an improved version whose statement and proof are independent of J, K and L. In Part III of this chapter, we compare our new version with the original J, K and L version and show that they are easily interderivable. §1. The Weak Double Recursion Theorem. Consider two r.e. relations, M1(x,y, z) and M2(x, y, z). For any number b, the relation M1(x,y,b) is an r.e. relation between x and y, and so by the weak recursion theorem (Theorem 1, Chapter 8), there is a number a such that ωa = x : M1(x, a, b). Likewise, for any number a, the relation M2(x,a,y) is an r.e. relation between x and y, and so there is a number b such that ωb, = x : M2(x,a,b). Our next theorem tells us that we can choose a and b so that both these conditions hold simultaneously.

Shepherdson Revisited

Theorems A and B of the last chapter were proved using previous results about generativity, Kleene pairs, complete effective inseparability, and double generativity. Yet the two theorems made no mention of these notions; they referred only to the notions of universality, double universality and semi-double universality. [These three notions, by the way, unlike the four notions mentioned above, were denned without reference to any indexing; they are what we would call index-free.] Is it not possible to give more direct proofs of Theorems A and B that do not require all the antecedent machinery of Chapters 4 and 5? We are about to show that it is possible; we will simply transfer Shepherdson’s arguments about first-order systems to recursion theory itself. We shall prove some “recursive function-theoretic” analogues of Shepherdson’s theorems which will provide new proofs of Theorems A and B (in fact, a strengthening of Theorem A will result).

Double Generativity and Complete Effective Inseparability

§1. Complete Effective Inseparability. A disjoint pair (A1,A2) is by definition recursively inseparable if no recursive superset of A1 is disjoint from A2. This is equivalent to saying that for any disjoint r.e. supersets ωi and ωj of A1 and A2, the set ωi is not the complement of ωj —in other words, there is a number n outside both ωi and ωj. The disjoint pair (A1,A2) is called effectively inseparable—abbreviated E.I.—if there is a recursive function δ(x, y)—called an E.I. function for (A1, A2)—such that for any numbers i and j such that A1⊆ ωi and A2Í ωj. with ωi being disjoint from ωj, the number d(i , j) is outside both a;,- and ωj. We shall call a disjoint pair (A1, A2) completely E.I. if there is a recursive function δ(x, y)—which we call a complete E.I. function for (A1, A2)—such that for any numbers i and j, if A1⊆ ωi and A2Í ωj, then δ(i , j) Í ωi ↔ d(i , j)Í ωj (in other words, d(i, j) is either inside or outside both sets ωi and ωj.). [If ωi and ωj happen to be disjoint, then, of course, d(i, j) is outside both ωi and ωj, so any complete E.I. function for (A1,A2) is also an E.I. function for (A1,A2) In a later chapter, we will prove the non-trivial fact that if (A1, A2) is E.I. and A1 and A2 are both r.e., then (A1,A2) is completely E.I. [The proof of this uses the result known as the double recursion theorem, which we will study in Chapter 9.] Effective inseparability has been well studied in the literature. Complete effective inseparability will play a more prominent role in this volume—especially in the next few chapters. Proposition 1. (1) If (A1,A2) is completely E.I., then so is (A2,A1) —in fact, if d(x,y) is a complete E.I. function for (A1,A2), then d(y,x) is a complete E.I. function for (A2, A1).

Generative Sets and Creative Systems

we now have the background to study the beautiful subject of creative and productive sets inaugurated by Emil Post [1944]. This plays a key rôle in the metamathematical study of incompleteness and undecidability. §1. Productive and Creative Sets. To say that a set A is not r.e. is equivalent to saying that for any r.e. subset ωi of A, there is a number in A not in ωi . A is called productive if there is a recursive function f(x)—called a productive function for A—such that for any number i, if ωi Í A, then f(i) Î A - ωi . [Informally, this means that it is not only true that no r.e. subset of A is A, but given any such r.e. subset, we can effectively find a number which is in A but not in the subset.] A set A is called creative (after Post) if it is r.e., and its complement is productive. Let us note that a productive function for the complement of a set A is a recursive function f(x) such that for any number i such that ωi is disjoint from A, the number f(i) lies outside both ωi and A. A system S is called productive if the set P of Gödel numbers of the provable formulas of S is a productive set; S is called creative if the set P is creative. As we will see, the complete theory N is not only not axiomatizable but is productive, and the system P.A. is not only undecidable but creative. Post’s Sets C and K. A simple example of a creative set is Post’s set C —the set of all numbers x such that x Î ωx. Thus for any number i, i Î C ↔ i Î ωi. If ωi is disjoint from C, then i is outside both ωi and C and, therefore, the identity function I(x) is a productive function for C͂. Therefore, C͂ is productive, and since C is obviously r.e., C is creative.

Prerequisites

As we remarked in the preface, although this volume is a sequel to our earlier volume G.I.T. (Gödel’s Incompleteness Theorems), it can be read independently by those readers familiar with at least one proof of Gödel’s first incompleteness theorem. In this chapter we give the notation, terminology and main results of G.I.T. that are needed for this volume. Readers familiar with G.I.T. can skip this chapter or perhaps glance through it briefly as a refresher. §0. Preliminaries. we assume the reader to be familiar with the basic notions of first-order logic—the logical connectives, quantifiers, terms, formulas, free and bound occurrences of variables, the notion of interpretations (or models), truth under an interpretation, logical validity (truth under all interpretations), provability (in some complete system of first-order logic with identity) and its equivalence to logical validity (Gödel’s completeness theorem). we let S be a system (theory) couched in the language of first-order logic with identity and with predicate and/or function symbols and with names for the natural numbers. A system S is usually presented by taking some standard axiomatization of first-order logic with identity and adding other axioms called the non-logical axioms of S.we associate with each natural number n an expression n̅ of S called the numeral designating n (or the name of n).we could, for example, take 0̅,1̅,2̅, . . . ,to be the expressions 0,0', 0",..., as we did in G.I.T. we have our individual variables arranged in some fixed infinite sequence v1, v2,..., vn , . . . . By F(v1, ..., vn) we mean any formula whose free variables are all among v1,... ,vn, and for any (natural) numbers k1,...,kn by F(к̅1 ,... к̅n), we mean the result of substituting the numerals к̅1 ,... к̅n, for all free occurrences of v1,... ,vn in F respectively.

Recursion Theorems

We have proved that the complement of every completely productive set (in other words, every generative set) is universal, and this was enough to establish Theorem A of Chapter 6. In Chapter 10 we will prove Myhill’s stronger result that the complement of every productive set is universal. For this proof, we will need the recursion theorem of this chapter. Recursion theorems (which can be stated in many forms) have profound applications in recursive function theory and metamathematics, and we shall devote considerable space to their study. To illustrate their rather startling nature, consider the following mathematical “believe-it-or-not’s”: Which of the following propositions, if true, would surprise you? . . . 1. There is a number n such that ωn = ωn+1.

Universal and Doubly Universal Systems

We now turn to two theorems (Theorems A and B below) that will play a major role in this study. We will give three different proofs of them in the course of this volume, since each proof reveals certain interesting features of its own. . . . Theorem A. If( A1 , A2 ) is semi-D.U. and A1 and A2 are both r.e., then A1 and A2 are both universal sets. Theorem B. I f ( A1 , A2 ) is semi-D.U. and A1 and A2 are both r.e., then (A1, A2) is D.U. . . Of course, Theorem A is a trivial corollary of Theorem B, but our proofs of Theorem A reveal facts not revealed by our proofs of Theorem B. We give our first proofs in this chapter. After each proof, we establish a metamathematical corollary: Theorem A yields the result of Ehrenfeucht-Feferman [1960] that for any consistent axiomatizable Rosser system S for sets in which all recursive functions of one argument are strongly definable, all r.e. sets are representable in S. Theorem B yields the stronger result of Putnam-Smullyan [1960]— that any such system S is an exact Rosser system for sets. This result is apparently incomparable in strength with Shepherdson’s result that any consistent axiomatizable Rosser system for binary relations is an exact Rosser system for sets. Both results, of course, yield different proofs that every consistent axiomatizable extension of (R) is an exact Rosser system for sets. §1. Generativity and Universality. We have shown that every universal set is generative. Our first proof of Theorem A will be based on the converse. Theorem 1. Every generative set is universal. We will, in fact, prove something considerably stronger which will have other applications as well. Consider a collection C of r.e. sets.