Recursive Enumerability and Recursivity

Abstract

Abstract Having proved that Peano Arithmetic is incomplete, we can ask another question about the system. Is there any algorithm (mechanical procedure) by which we can determine which sentences are provable in the system and which are not? This brings us to the subject of recursive function theory, to which we now turn. we are denning a relation (or set) to be r.e. (recursively enumerable) iff it is Σ1, and to be recursive iff it and its complement are r.e. An equivalent definition of recursive enumerability is represent ability in some finitely axiomatizable system (as we will prove). Many other characterizations of recursive enumerability and recursivity can be found in the literature (cf., e.g., Kleene [1952], Turing [1936], Post [1944], Smullyan [1961], Markov [1961]), but the Σ1-characterization fits in best with the overall plan of this volume. The fact that so many different and independently formulated definitions turn out to be equivalent adds support to a thesis proposed by Church—namely that any function that is effectively calculable in the intuitive sense is a recursive function. Interesting discussions of Church’s thesis can be found in Kleene [1952] and Rogers [1967]. In this chapter, we establish a few basic properties of recursive enumerability that will be needed in just about all the chapters that follow. §1. Some Closure Properties. It will be convenient to regard sets as special cases of relations (sets are thus relations of one argument or relations of degree 1). It will be convenient to use the l-notation “λx1,...,xn : (...)”, read “the set of all n-tuples (x1,..., xn) such that (...)”. For example, for any relation λ(x1, x2, x3), the relation λx1x2x3: R(x2 x2, x3) is the set of all triples (x1,x2,x3) (of natural numbers) such that R(x2,x2,x1) holds. we sometimes write “x: (. . . )” for “λx: ( . . . ) ”.