MANY - ONE REDUCIBILITY WITHIN THE TURING DEGREES OF THE HYPERARITHMETIC SETS HJx ) ( i )

Spector [l3] has proven that the hyperarithmetic sets H Ax) and H Ax) have the same Turing degree iff \a\ = |è|. Y. Moschovakis has bat the sets /_a proven that H {x) under many-one reducibility for \a\ = y and a e Q have nontrivial reducibility properties if y is not of the form a + 1 or a + cu for any ordinal a. In particular, he proves that there are chains of order type o and incomparable many-one degrees within these Turing degrees. In Chapter II, we extend this result to show that any countable partially ordered set can be embedded in the many-one degrees within these Turing degrees. 2 In Chapter III, we prove that if y is also not of the form a + o> for some ordinal a, then there is no minimal many-one degree of the form H (x) '" this Turing degree, answering a question of Y. Moschovakis posed in [8J. In fact, we prove that given H W there are H Ax) and H (x) both many-one reducible to H (x) with incomparable many-one degrees, \a\ \b\ = Ici = y. a CHAPTER I. PRELIMINARIES For the most part we adopt the terminology and notation as introduced by Kleene in [4], u], and [6J. For definiteness, our Go'del numbering of the partial recursive functions of «-variables will be the particular one given in [4]. Also, we use freely the function U(z) and the T-predicates of Kleene [4]. We assume familiarity with the notions of many-one (one-one) reducibility of A to B and denote this by A <m B (A <1 B) [10] and til]. Similarly, we write A <T B if A is Turing reducible to B [7], i.e., A is recursive in B [5]. Degrees will refer to the equivalence classes of sets indistinguishable under a specified Presented to the Society, September 23, 1968 under the title Reducibility orderings of the hyperarithmetic predicates; received by the editors June 25, 1971 and, in revised form, June 1, 1973. AMS (MOS) subject classifications (1970). Primary 02F27, 02F35.