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arXiv 1310.0891math.LO
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Prikry型强迫和最小$α$度

Prikry-type forcing and minimal $α$-degree

Yang Sen

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摘要：

在本文中，我们介绍了几类Prikry型强制概念，其中两个是用来产生最小的一般扩张，第三个是应用在$α$ -递归理论，以产生最小的覆盖。第一个强制作为预热，在可测量基数（在$V$中）上产生最小通用扩展，第二个强制在$ω$ -可测量基数的极限$\langleγ_n\colon n<ω\rangle$，使得每个$γ_n$（$n>0$）携带$γ_{n-1}$ -许多正常测量。通过一个$V_γ$ -度的概念（见定义\ref{def:vgammadegree}），我们将最小类属扩张的第二个Prikry型构造转化为$α$ -递归理论中的最小度构造.更明确地说，\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*}

英文摘要：

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*}

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