Hyperarithmetical Index Sets in Recursion Theory

We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. An extension yields a nj-complete index set. We also classify the index set of quasimaximal sets, of coinfinite r.e. sets not having an atomless superset, and of r.e. sets major in a fixed nonrecursive r.e. set. 0. Introduction. The present paper deals with index sets, i.e., sets of indices of partial recursive (p.r.) functions and recursively enumerable (r.e.) sets that are defined through the p.r. functions or r.e. sets they code. The early results in index sets used geometric arguments in oneor two-dimensional arrays: Rogers showed the £3 and n3-completeness of the index sets of recursive and simple sets, respectively, in a finite injury argument. Lachlan, D. A. Martin, R. W. Robinson, and Yates (1968, unpublished, later appearing in Tulloss [Tu71]) showed the ^-completeness of the index set of maximal sets in an infinite injury argument. Tulloss [ibid.] also mentions for the first time the question whether the index set of quasimaximal sets is E5-complete. However, the geometric method was too complex at higher levels of the arithmetical hierarchy. During the 1970's, progress in index sets was mainly made in other areas by several Russian mathematicians as well as L. Hay. Schwarz [Schta] was the first to introduce induction into index set proofs (in the r.e. degrees) and was able to show that the index sets of low„ and highn r.e. sets are £„+3 and En+4-complete, respectively. Solovay [JLSSta] then extended Schwarz's methods to show the £w+i-completeness of the index sets of low<w (lown for some n) and of high<w (high„ for some n) r.e. sets as well as the H^+i-completeness of the index set of intermediate degrees (degrees neither low<w nor high<w). In this paper, we exhibit a family of algebraically invariant properties LWliWdefinable in £, that yields index sets at any level of the %perarithmetical hierarchy. The proof is based on induction and Lachlan's theorem [La68] that any E3-Boolean algebra is isomorphic to the lattice of r.e. supersets of some r.e. set (modulo finite sets). It uses tree arguments and the fact that the Cantor-Bendixson rank of a tree corresponds to certain properties of the lattice of r.e. supersets of the set constructed. An extension yields a nj-complete index set. A corollary shows the ^-completeness of the index set of quasimaximal sets, thereby settling this longopen question. Further results classify the index sets of atomic sets and of r.e. sets major in a fixed nonrecursive r.e. set. Our notation is fairly standard and generally follows Soare's forthcoming book Recursively Enumerable Sets and Degrees [Sota]. Received by the editors July 7, 1986 and, in revised form, November 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03D25. This paper is an extended version of part of the author's thesis. He wishes to thank his thesis advisor, R. I. Soare, as well as T. A. Slaman and J. Steel for helpful suggestions and comments. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page