论文标题
可数可允许的集合的部分优先末端扩展
Partially-elementary end extensions of countable admissible sets
论文作者
论文摘要
Kaufmann的结果表明，如果$L_α$是可计数，可接受的，并且满足$π_n\ textsf {-collection} $，则$ \ langlel_α，\ in \ rangle $具有适当的$σ_{n+1} $ - 基本结束。本文研究了$ \ langlel_α，\ in \ rangle $中所持的理论的程度，可以将其转移到Kaufmann结果保证的部分元素末端扩展中。我们表明，有$l_α$满足完整的分离，powerset和$π_n\ textsf {-collection} $没有适当的$σ_{n+1} $ - 基本末端扩展，满足满足$π_{n} n} \ textsf {-collection} $或$ fextsf}相比之下，我们表明，如果$ a $是可计的可允许的集合，满足$π_n\ textsf {-collection} $ and $ t $是一种递归枚举的理论，它在$ \ langle a，in \ rangle $，然后是$ \ langle a in \ langle a in \ rangle $ $ $ $ $ -
A result of Kaufmann shows that if $L_α$ is countable, admissible and satisfies $Π_n\textsf{-Collection}$, then $\langle L_α, \in \rangle$ has a proper $Σ_{n+1}$-elementary end extension. This paper investigates to what extent the theory that holds in $\langle L_α, \in \rangle$ can be transferred to the partially-elementary end extensions guaranteed by Kaufmann's result. We show that there are $L_α$ satisfying full separation, powerset and $Π_n\textsf{-Collection}$ that have no proper $Σ_{n+1}$-elementary end extension satisfying either $Π_{n}\textsf{-Collection}$ or $Π_{n+3}\textsf{-Foundation}$. In contrast, we show that if $A$ is a countable admissible set that satisfies $Π_n\textsf{-Collection}$ and $T$ is a recursively enumerable theory that holds in $\langle A, \in \rangle$, then $\langle A, \in \rangle$ has a proper $Σ_n$-elementary end extension that satisfies $T$.
