Cone avoiding closed sets
Cone avoiding closed sets
复制标题
圆锥避免闭集
DOI：
10.1090/s0002-9947-2014-06049-2
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Lu Liu
中科院分区：
文献类型：
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作者：
Lu Liu
我们证明了：对于$2^{&t；\omega}$的任意子树$T$且每个元素可扩展到一条路径，给定的可数类$\mathcal{M}$在不交并下闭合，以及任何集合$A$，如果$\mathcal{M}$的任何成员都不对任何$k$-枚举$T$，则存在一个包含在$A$或$\bar{A}$中的无限集，使得对于每个在数学{M}$中的$C\，$C\Oplus G$也不是强$k$-枚举$T$.我们给出了这一结果的应用，其中包括：(1)$\mathsf{RT_2^2}$不蕴含$\mathsf{WWKL_0}$；(2)(Ambos-Spies等人2004)$\mathsf{dnr}$严格弱于$\mathsf{WWKL_0}$；(3)(Kjos-Hanssen 2009)对任何Martin-L随机集$A$A$或$\bar{A}$包含一个不计算任何Martin-L{o}f随机集的无限子集；等。我们还讨论了这一结果的进一步推广。
We prove that for an arbitrary subtree $T$ of $2^{<\omega}$ with each element extendable to a path, a given countable class $\mathcal{M}$ closed under disjoint union, and any set $A$, if none of the members of $\mathcal{M}$ strongly $k$-enumerate $T$ for any $k$, then there exists an infinite set contained in either $A$ or $\bar{A}$ such that for every $C\in\mathcal{M}$, $C\oplus G$ also does not strongly $k$-enumerate $T$. We give applications of this result, which include: (1) $\mathsf{RT_2^2}$ doesn't imply $\mathsf{WWKL_0}$; (2) (Ambos-Spies et al.2004) $\mathsf{DNR}$ is strictly weaker than $\mathsf{WWKL_0}$; (3) (Kjos-Hanssen 2009) for any Martin-L\"{o}f random set $A$ either $A$ or $\bar{A}$ contains an infinite subset that does not compute any Martin-L\"{o}f random set; etc. We also discuss further generalizations of this result.