使用泛型和随机数连接的限制
Limits to joining with generics and randoms
摘要：
Posner和Robinson（1981）证明了如果$S \subseteq ω$是不可计算的，则存在一个$G \subseteq ω$使得$S \oplus G \geq_T G'$。Shore和Slaman（1999）将这一结果推广到所有的$n \in ω$，表明如果$S \nleq_T \emptyset^{(n-1)}$，则存在一个$G$，使得$S \oplus G \geq_T G^{(n)}$。他们的论证使用了Kumabe-Slaman强迫，因此他们得到的集合与波斯纳-罗宾逊定理不同，在任何方面都不是科恩强迫的通用集合。我们回答的问题，这是否是一个必要的复杂性表明，对于所有的$n \geq 1$，设置$G$的海岸，Slaman定理不能被选择为甚至弱2-通用。我们的结果适用于其他几个有效的强制概念中常用的可计算性理论，我们还证明了集$G$不能被选为2-随机。
英文摘要：
Posner and Robinson (1981) proved that if $S \subseteq ω$ is non-computable, then there exists a $G \subseteq ω$ such that $S \oplus G \geq_T G'$. Shore and Slaman (1999) extended this result to all $n \in ω$, by showing that if $S \nleq_T \emptyset^{(n-1)}$ then there exists a $G$ such that $S \oplus G \geq_T G^{(n)}$. Their argument employs Kumabe-Slaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing in any way. We answer the question of whether this is a necessary complication by showing that for all $n \geq 1$, the set $G$ of the Shore-Slaman theorem cannot be chosen to be even weakly 2-generic. Our result applies to several other effective forcing notions commonly used in computability theory, and we also prove that the set $G$ cannot be chosen to be 2-random.
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