Prikry-type forcing and minimal $\alpha$-degree

Abstract

In this paper, we introduce several classes of Prikry-type forcing notions, two of which are used to produce minimal generic extensions, and the third is applied in $\alpha$-recursion theory to produce minimal covers. The first forcing as a warm up yields a minimal generic extension at a measurable cardinal (in $V$), the second at an $\omega$-limit of measurable cardinals $\langle\gamma_n\colon n<\omega\rangle$ such that each $\gamma_n$ ($n>0$) carries $\gamma_{n-1}$-many normal measures. Via a notion of $V_\gamma $-degree (see Definition \ref{def:vgammadegree}), we transfer the second Prikry-type construction for minimal generic extensions to a construction for minimal degrees in $\alpha$-recursion theory. More explicitly, \begin{theorem*} Suppose $\langle\gamma_n\colon n<\omega\rangle$ is a strictly increasing sequence of measurable cardinals such that for each $n>0$, $\gamma_n$ carries at least $\gamma_{n-1}$-many normal measures. Let $\gamma=\sup\{\gamma_n\colon n<\omega\}$. %Then for each $n$, $\gamma$ is $\Sigma_n$-admissible. Then there is an $A\subset\gamma$ such that \begin{itemize} \item[(a)] $(L_{\gamma},\in,A)$ is not admissible. \item[(b)] The $\gamma$-degree that contains $A$ has a minimal cover. \end{itemize} \end{theorem*}