Many-one reducibility within the Turing degrees of the hyperarithmetic sets $H_{a}(x)$

Abstract:

Spector [13] has proven that the hyperarithmetic sets ${H_a}(x)$ and ${H_b}(x)$ have the same Turing degree iff $|a| = |b|$. Y. Moschovakis has proven that the sets ${H_a}(x)$ under many-one reducibility for $|a| = \gamma$ and $a \in \mathcal {O}$ have nontrivial reducibility properties if $\gamma$ is not of the form $\alpha + 1$ or $\alpha + \omega$ for any ordinal a. In particular, he proves that there are chains of order type ${\omega _1}$ 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 $\gamma$ is also not of the form $\alpha + {\omega ^2}$ for some ordinal a, then there is no minimal many-one degree of the form ${H_a}(x)$ in this Turing degree, answering a question of Y. Moschovakis posed in [8]. In fact, we prove that given ${H_a}(x)$ there are ${H_b}(x)$ and ${H_c}(x)$ both many-one reducible to ${H_a}(x)$ with incomparable many-one degrees, $|a| = |b| = |c| = \gamma$.