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).