The priority method for the construction of recursively enumerable sets

Abstract

For each i < ~ let C(i) c p(~) the power set of ~. Suppose that an r.e. set A (N{C(i) I i < ~} is found by a construction which appears to succeed because the need for A (C(0) is viewed as having first importance, the need for A (C(1) is viewed as having second importance, and so on, then one says that the construction uses the method. Beyond this, apart from providing a long list of examples, it is difficult to be more precise about What the priority method is. Discussions about the general nature of the method can be found in [8], [k], and [i]. Historically, the term arose from the way in which Friedberg [2] and Mu~nik [5] independently solved Post's problem. In his review [7] of [6] Myhill remarked that Friedberg's method seemed to be an effective analogue of Baire's theorem from topology. Below we shall develop this analogy of Myhill's for three different constructions: (i) a way of solving Post's problem due to Sacks, (2) the construction of a minimal pair of r.e. degrees achieved independently by Yates and the author, and (3) the density theorem for r.e. degrees again due to Sacks. In each case we shall isolate the way in which A (C(i) is ensured for a particular i < ~. Most of our notation is standard. N denotes the set of natural numbers, P(S) denotes the power set of S, Pfin(S) denotes the set of finite subsets of S. For any set A, c A denotes the characteristic function of A. L(x) denotes {YlY~N~Y<X}. If ~ is af~ction~d E aset FIR denotes the restriction of F to E; if E is L(x) we write FIx for FIL(x). We use -<T to denote the relation of Turing reducibility and <T for proper Turing reducibility. i. Baire's theorem. For the purposes of the analogy that we wish to draw below we shall take Baire's theorem to be the one which states that in a compact Hausdorff space X, if X C(i) is nowhere dense for each i < ~, then N{C(i) I i < c0} # ~. This can be broken down as follows. Call a closed set