Prikry-type forcing and minimal $α$-degree
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 $α$-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 $ω$-limit of measurable cardinals $\langleγ_n\colon n<ω\rangle$ such that each $γ_n$ ($n>0$) carries $γ_{n-1}$-many normal measures. Via a notion of $V_γ$-degree (see Definition \ref{def:vgammadegree}), we transfer the second Prikry-type construction for minimal generic extensions to a construction for minimal degrees in $α$-recursion theory. More explicitly, \begin{theorem*} Suppose $\langleγ_n\colon n<ω\rangle$ is a strictly increasing sequence of measurable cardinals such that for each $n>0$, $γ_n$ carries at least $γ_{n-1}$-many normal measures. Let $γ=\sup\{γ_n\colon n<ω\}$. %Then for each $n$, $γ$ is $Σ_n$-admissible. Then there is an $A\subsetγ$ such that \begin{itemize} \item[(a)] $(L_γ,\in,A)$ is not admissible. \item[(b)] The $γ$-degree that contains $A$ has a minimal cover. \end{itemize} \end{theorem*}