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Spector [13] has proven that the hyperarithmetic sets H a(x) and H b(x) have the same Turing degree iff jai = |b|. Y. Moschovakis has proven that the sets H (x) under manyone reducibility for lal = y and a E 0 have nontrivial reducibility properties if y is not of the form a + 1 or a + co for any ordinal a. In particular, he proves that there are chains of order type co 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. In Chapter III, we prove that if y is also not of the form a + co2 for some ordinal a, then there is no minimal many-one degree of the form HP(x) in this Turing degree, answering a question of Y. Moschovakis posed in [81 In fact, we prove that given H (x) there are H b(x) and H (x) both many-one reducible to H (x) with incomparable many-one degrees, Ial = Ib| = ic| = Y. a CHAPTER I. PRELIMINARIES For the most part we adopt the terminology and notation as introduced by Kleene in [4], [5], and [6]. For definiteness, our G6del numbering of the partial recursive functions of n-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 [11]. 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.

Many-one reducibility within the Turing degrees of the hyperarithmetic sets 𝐻ₐ(𝑥)